Table Of ContentTRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S
WEIGHTED ORBITAL INTEGRALS
ZONGBIN CHEN
7
1
0
2
n Abstract. We explain an algorithm to calculate Arthur’s weighted orbital integral in
a
termsofthenumberofrationalpointsonthefundamentaldomainoftheassociatedaffine
J
Springer fiber. The strategy is to count the number of rational points of the truncated
2
affine Springerfibersin two ways: bytheArthur-Kottwitzreduction and by theHarder-
1
Narasimhanreduction. Acomparisonofresultsobtainedfromthesetwoapproachesgives
usrecurrencerelationsbetweenthenumberofrationalpointsonthefundamentaldomains
]
G of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we
calculate Arthur’sweighted orbital integrals for thegroup GL and GL .
A 2 3
.
h
t 1. Introduction
a
m Let k = F be the finite field with q elements. Let F = k((ǫ)) be the field of Laurent
q
[ series with coefficients in k, O = k[[ǫ]] the ring of integers of F, p = ǫk[[ǫ]] the maximal ideal
of O. We fix an algebraic closure k¯ of k, and also a compatible algebraic closure F of F.
1
×
v Let val :F → Q be the discrete valuation normalized by val(ǫ) = 1.
2
Let G be a connected split reductive group scheme over O, assume that char(k) > r :=
0
2 rk(G), where rk(G) is the semisimple rank of G. Let T be a maximal torus of G over O.
3 We make the assumption that the splitting field of T(F) is totally ramified over F. Let
0 S ⊂ TbethemaximalsplitsubtorusofToverO,letM0 = ZG(S). Weuseplainlettersfor
.
1 the special fiber of the corresponding group scheme, for example, G := G× Spec(k);
Spec(O)
0
and we use Gothic letters for the associated Lie algebras. We fix Haar measures dg and dt
7
on G(F) and T(F) respectively, normalized by dg(G(O)) = dt(T(O)) = 1.
1
: Let γ ∈ t(O) be a regular element, it is anisotropic in m (F). In this work, we are
v 0
i interested in Arthur’s weighted orbital integral
X
dg
r (1.1) J (γ) = J (γ, )= Ad(g)−1γ v (g) ,
a M0 M0 1g(O) ZT(F)\G(F)1g(O) M0 dt
(cid:0) (cid:1)
where isthecharacteristic functionofthelattice g(O)ing(F), andv (g)isArthur’s
1g(O) M0
weightfactor. Oneofourmainresultsstatesthatitcanbeexpressedintermsofthenumber
of rational points of the fundamental domain F of the affine Springer fiber X . The main
γ γ
idea is to count the number of rational points of the truncated affine Springer fibers in two
different ways: by the Arthur-Kottwitz reduction and by the Harder-Narasimhan reduction.
Before entering the details of our approach, we give examples of results that can be
obtained in this way. The calculations for the group G = GL is easy, the results are
2
1
2 ZONGBINCHEN
summarized in theorem 5.1, 5.2. But for the group G = GL , the calculations are already
3
quite non-trivial. There are 3 cases to deal with: the element γ can be split, mixed or
anisotropic. In all these cases, we are able to calculate |F (F )|, and to deduce Arthur’s
γ q
weighted orbital integral from it. When γ is split, it can always be conjugate by the Weyl
group such that
val(α (γ)) = val(α (γ)) ≤ val(α (γ)).
12 13 23
We call (n ,n ) = (val(α (γ)),val(α (γ))) the root valuation of γ.
1 2 12 23
Theorem 1.1. Let G = GL , T the maximal torus of diagonal matrices. Let γ ∈ t(O) be a
3
regular element with root valuation (n ,n ) ∈ N2, with n ≤ n . Arthur’s weighted orbital
1 2 1 2
integral for γ equals
n1 2n1+n2−1
J (γ) = vol(a /X (T))· i(q2i−1+q2i−2)+ (4n +2n −4i−3)qi
T T ∗ 1 2
"
Xi=1 i=Xn1+n2
+(n2+2n n )q2n1+n2 .
1 1 2
#
When γ is mixed, i.e. the splitting field of γ is isomorphic to F× ×k((ǫ12))×, it can be
conjugate to a matrix of the form
a
(1.2) γ = b .
bǫ
Let m = val(a), n = val(b), we have
Theorem 1.2. Let G = GL , let γ be a matrix in the form (1.2). When val(a) = m ≤ n,
3
we have
2m+n−1
J (γ) = vol(a /X (M ))· 2mq2m+n + 2(j −m−n)qj
M0 M0 ∗ 0
"
j=m+n+1
X
2m−1
j
− +1 qj .
2
#
j=0 (cid:18)(cid:22) (cid:23) (cid:19)
X
When val(a) = m > n, we have
3n
J (γ) = vol(a /X (M ))· (2n+1)q3n+1+ (2j −4n−1)qj
M M0 ∗ 0
"
j=2n+1
X
2n
j
− +1 qj .
2
#
j=0(cid:18)(cid:22) (cid:23) (cid:19)
X
TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS 3
When γ is anisotropic, Arthur’s weighted orbital integral is just the orbital integral, and
the result was essentially obtained by Goresky, Kottwitz and MacPherson [GKM2]. See
theorem 8.1, 8.2 for the counting result.
Nowlet meexplain our approach tothe calculation of Arthur’s weighted orbital integrals
using the geometry of the affine Springer fibers. The affine Springer fiber X is the closed
γ
sub scheme of the affine grassmannian X = G(F)/G(O) defined by the equation
X = g ∈ G(F)/G(O) Ad(g−1)γ ∈ g(O) .
γ
To simplify notations, we wr(cid:8)ite K = G(O). (cid:12)They can be used(cid:9)to geometrize Arthur’s
(cid:12)
weighted orbital integrals. The group T(F) acts on X by left translation. Let Λ =
γ
π (T(F)), we can show that
0
dg
Ad(g)−1γ v (g) = v (g),
ZT(F)\G(F)1g(O)(cid:0) (cid:1) M0 dt [g]∈ΛX\Xγ(Fq) M0
where [g] denotes the point gK ∈ X.
But this expression doesn’t facilitate the calculations of Arthur’s weighted orbital inte-
gral. We have to proceed in an indirect way. Let ξ ∈ aG be a generic element, Laumon
M0
and Chaudouard [CL2] introduce a variant of the weighted orbital integral
dg
(1.3) Jξ (γ) = Jξ (γ, )= Ad(g)−1γ wξ (g) ,
M0 M0 1g(O) ZT(F)\G(F)1g(O) M0 dt
(cid:0) (cid:1)
ξ
with a slightly different weight factor w (g). Although the two weight factors are not the
M0
same, Laumon and Chaudouard show that
J (γ) = vol(a /X (M ))·Jξ (γ).
M0 M0 ∗ 0 M0
ξ
The variant J (γ) has a better geometric interpretation. In fact, we can introduce a
notion of ξ-stabiMlit0y on the affine Springer fiber X (when T splits, this is done in [C2]),
γ
and show that
Jξ (γ) = |Xξ(F )|.
M0 γ q
The advantage of this variant is clear: it is a plain count rather than a weighted count.
Moreover, we can use the Harder-Narasimhan reduction to get |Xξ(F )| recursively from
γ q
|X (F )|, if only the latter is finite. Unfortunately, this is not the case, as can be seen from
γ q
the fact that the free abelian group Λ acts freely on X .
γ
Let Π be a positive (G,M )-orthogonal family, we can introduce a truncation X (Π) to
0 γ
overcome the finiteness issue. When Π is sufficiently regular, we can reduce the calculation
of the rational points on X (Π) to that of the fundamental domain F , by the Arthur-
γ γ
Kottwitz reduction. Recallthatthefundamental domainF isintroducedin[C3]toplaythe
γ
roleofanirreduciblecomponentofX (AlltheirreduciblecomponentsofX areisomorphic
γ γ
because Λ acts simply and transitively on them). The Arthur-Kottwitz reduction is a
construction that decompose X (Π) into locally closed sub varieties, which are iterated
γ
affine fibrations over the fundamental domains FM for the Levi subgroups M containing
γ
4 ZONGBINCHEN
M . The counting result is summarized in theorem 3.4. In particular, it shows that X (Π)
0 γ
depends quasi-polynomially on the truncation parameter.
On the other hand, the Harder-Narasimhan reduction doesn’t behave well on X (Π). In
γ
fact, near the boundary, the Harder-Narasimhan strata are generally not affine fibrations
over truncations of XM,ξM for the Levi subgroups M. To overcome this difficulty, we cut
γ
X (Π) into two parts: the tail and the main body. Roughly speaking, the tail is the union
γ
of the “boundary irreducible components” of X (Π), and the main body is its complement.
γ
The Harder-Narasimhan reduction works well on the main body, and we can useit to count
thenumber of rational points. The resultis summarized in theorem 4.7, it canbeexpressed
in terms of |XM,ξM(F )|, M being the Levi subgroups of G containing M . The counting
γ q 0
points on the tail is more complicated, it spreads over §4.2.2, §4.2.3, §4.2.4. In principle,
we can use the Arthur-Kottwitz reduction iteratively to express it in terms of |FM(F )|’s.
γ q
But we are not able to obtain an explicit expression, what we get is only a recursion.
These two different approaches to counting rational points on X (Π) give us a recursive
γ
equation that involves the |FM(F )|’s and the |XM,ξM(F )|’s. Solving it, we can express
γ q γ q
the latter ones in terms of the former ones. The problem of calculating Arthur’s weighted
orbitalintegralsisthusreducedtocountingpointsonF . Inourpointofview,thegeometry
γ
of F is simpler than that of Xξ. Goresky, Kottwitz and MacPhersion have conjectured
γ γ
that the cohomological groups of X is pure in the sense of Deligne. As we have shown
γ
in [C3], this is equivalent to the cohomological purity of F . In fact, it is even expected
γ
that F admits a Hessenberg paving. When the torus T splits, we [C4] make a conjecture
γ
on the Poincaré polynomial of F , assuming the cohomological purity of F . This gives a
γ γ
conjectural expression for |F (F )|. We reproduce it here for the convenience of the reader.
γ q
According to Chaudouard-Laumon [CL1], the homology of the truncated affine Springer
fibers can be expressed in terms of its 1-skeleton under the T-action, whenever they are
cohomologically pure. We can adapt their result to the fundamental domain F . The torus
γ
T acts on F with finitely many fixed points FT, but the union FT,1 of the 1-dimensional
γ γ γ
T,1
T-orbits is not discrete. The bigger torus T = T × G acts on F with finitely many
m γ
e
T,1
1-dimensional T-orbits, let F be their union. The result of [CL1] states that there is an
γ
e
exact sequence
e 0 → H∗(F ) → H∗(FTe)→ H∗(FTe,1,FTe),
Te γ Te γ Te
and there is an isomorphism
H∗(F ) = H∗(F )⊗ Q ,
γ Te γ HT∗e(pt) l
where all the cohomology groups take coefficients in Q . Let Γ be the graph with vertices
l
FTe and with edges FTe,1. Two vertices are linked by an edge if and only if they lie on the
γ γ
closureof thecorresponding 1-dimensional T-orbit. Wecallitthemoment graph of F with
γ
respect to the action of T.
Let obea total order among theverticeseof the graph Γ, it willserve as the paving order.
We associate to it an aceyclic oriented graph (Γ,o) such that the source of each arrow is
TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS 5
greater than its target with respect to o. For v ∈ Γ, denote by no the number of arrows
v
having source v.
Definition 1.1. The formal Betti number bo associated to the order o is defined as
2i
bo = ♯{v ∈ Γ : no = i}.
2i v
We call
Po(t) = bo t2i
2i
i
X
the formal Poincaré polynomial associated to the order o.
Definition 1.2. For P (t), P (t)∈ Z[t], we say that P (t) < P (t) if the leading coefficient
1 2 1 2
of P (t)−P (t) is positive.
2 1
Conjecture 1.1. Let P(t) be the Poincaré polynomial of F , then
γ
P(t)= min{Po(t)},
o
where o runs through all the total orders among the vertices of Γ.
In particular, the conjecture implies that
|F (F )| = min{Po(q1/2)}.
γ q
o
Together withtherecurrence relationbetween |Xξ(F )|and|F (F )|, itgivesaconjectural
γ q γ q
complete answer tothedetermination ofArthur’s weighted orbital integral inthe splitcase.
Notations. We fix a split maximal torus A of G over k. Without loss of generality, we
suppose that A ⊂ M . Let Φ = Φ(G,A) be the root system of G with respect to A, let
0
W be the Weyl group of G with respect to A. For any subgroup H of G which is stable
under the conjugation of A, we note Φ(H,A) for the roots appearing in Lie(H). We fix a
Borel subgroup B of G containing A. Let ∆ = {α ,··· ,α } be the set of simple roots
0 1 r
with respect to B , let (̟ )r be the corresponding fundamental weights. To an element
0 i i=1
α ∈ ∆, we have a unique maximal parabolic subgroup P of G containing B such that
α 0
Φ(N ,A) ∩ ∆ = α, where N is the unipotent radical of P . This gives a bijective
Pα Pα α
correspondence between the simple roots in ∆ and the maximal parabolic subgroups of G
containing B . Any maximal parabolic subgroup P of G is conjugate to certain P by an
0 α
element w ∈ W, the element w̟ doesn’t depend on the choice of w, we denote it by ̟ .
α P
We use the (G,M) notation of Arthur. Let F(A) be the set of parabolic subgroups
of G containing A, let L(A) be the set of Levi subgroups of G containing A. For every
M ∈ L(A), we denote by P(M) the set of parabolic subgroups of G whose Levi factor is
M, and by F(M) the set of parabolic subgroups of G containing M. For P ∈ P(M), we
denote by P− the opposite of P with respect to M.
Let X∗(M) = Hom(M,G ) and X (M) = Hom(X∗(M),Z). Let a∗ = X∗(M)⊗R and
m ∗ M
a = X (M)⊗R. The restriction X∗(M) → X∗(A) induces an injection a∗ ֒→ a∗. Let
M ∗ M A
(aM)∗ be the subspace of a∗ generated by Φ(M,A). We have the decomposition in direct
A A
sums
a∗ = (aM)∗⊕a∗ .
A A M
6 ZONGBINCHEN
The canonical pairing X (A)×X∗(A) → Z can be extended bilinearly to a ×a∗ → R.
∗ A A
For M ∈L(A), we can embed a in a as the orthogonal subspace to (aM)∗. Let aM ⊂a
M A A A A
be the subspace orthogonal to a∗ . We have the dual decomposition
M
a = a ⊕aM,
A M A
letπ , πM betheprojectionstothetwofactors. Moregenerally, forL,M ∈ F(A), M ⊂ L,
M
we also have a decomposition
aG = aG ⊕aL .
M L M
To save notation, we also write π , πL for the projections to the two factors.
L
We identify X (A) with A(F)/A(O) by sending χ to χ(ǫ). With this identification, the
∗
canonical surjection A(F) → A(F)/A(O) can be viewed as
(1.4) A(F) → X (A).
∗
We use Λ to denote the quotient of X (A) by the coroot lattice of G (the subgroup of
G ∗
X (A) generated by the coroots of A in G). We have a canonical homomorphism
∗
(1.5) G(F) → Λ ,
G
which is characterised by the following properties: it is trivial on the image of G (F) in
sc
G(F) (G is the simply connected cover of the derived group of G), and its restriction to
sc
A(F) coincides with the composition of (1.4) with the projection of X (A) to Λ . Since
∗ G
the morphism (1.5) is trivial on G(O), it descends to a map
ν : X → Λ ,
G G
whose fibers are the connected components of X.
Finally, we suppose that γ ∈ t(O) satisfies γ ≡ 0 mod ǫ to avoid unnecessary complica-
tions.
Acknowledgements. We want to thank Gérard Laumon for the discussions which have
led to this work.
2. (Weighted) orbital integrals and the affine Springer fibers
We recall briefly the geometrization of the (weighted) orbital integrals using the affine
Springer fibers. We fix a regular element γ ∈ t(O) as in the introduction. Let P = M N
0 0 0
be the unique element in P(M ) which contains B .
0 0
2.1. Orbital integrals. Consider the orbital integral
dg
IG = Ad(g)−1γ .
γ ZT(F)\G(F)1g(O) dt
(cid:0) (cid:1)
TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS 7
Use the translation invariance of the Haar measure, we can rewrite it as
dg
IG = dt(Λ\T(F))−1 Ad(g)−1γ
γ ZΛ\G(F)1g(O) dt
(cid:0) (cid:1)
dg(G(O))
= Ad(g)−1γ
dt(T(O)) 1g(O)
g∈Λ\G(F)/G(O)
X (cid:0) (cid:1)
(2.1) = Λ\{g ∈ G(F)/G(O) | Ad(g)−1γ ∈g(O)}
The set at the end of the equ(cid:12)ation can be given a scheme structure ove(cid:12)r k. In fact, there is
(cid:12) (cid:12)
an ind-k-scheme X, called the affine grassmannian, which satisfies
X(Fqn) = G(Fqn((ǫ)))/G(Fqn[[ǫ]]), ∀n∈ N.
This can be written succinctly as X = G((ǫ))/G[[ǫ]]. The affine Springer fiber at γ is the
closed sub-ind-k-scheme of X defined by
X = g ∈ G((ǫ))/G[[ǫ]] Ad(g−1)γ ∈g[[ǫ]] .
γ
It is in fact a scheme over k of(cid:8)finite dimension(cid:12), but it is general(cid:9)ly not of finite type. The
(cid:12)
free discret abelian group Λ acts freely on it, and the quotient Λ\X is projective over k.
γ
The equation (2.1) can be written as
(2.2) IG = |Λ\X (F )|.
γ γ q
The calculation of IG can be reduced to that of IM0. Recall that for P = MN ∈ F(A),
γ γ
we have the retraction
f : X → XM
P
which sends [g] = gK to [m] := mM(O), where g =nmk, n∈ N(F), m ∈ M(F), k ∈ K is
theIwasawa decomposition. Wewant topoint outthat theretraction f is notamorphism
P
betweenind-k-schemes, butitsrestrictiontotheinverseimageofeachconnectedcomponent
of XM:
f : f−1(XM,ν) → XM,ν, ν ∈ Λ ,
P P M
is actually a morphism over k between ind-k-schemes.
More generally we can define fL : XL → XM for L ∈ L(A), L ⊃ M and P ∈ PL(M).
PL L
These retractions satisfythe transitivity property: Suppose that Q ∈ P(L) satisfies Q ⊃ P,
then
f = fL ◦f .
P P∩L Q
Now take P = MN ∈ F(M ), the retraction f sends X to XM. To see this, for
0 P γ γ
[g] ∈ X , let g = nmk be the Iwasawa decomposition as above. We can write g = mn′k
γ
with n′ = m−1nm ∈ N(F). The equation Ad(g)−1γ ∈ g(O) implies
Ad(m)−1γ ∈ Ad(n′)g(O)∩m(F)= m(O),
which means that f ([g]) = [m] ∈ XM.
P γ
8 ZONGBINCHEN
Proposition 2.1 (Kazhdan-Lusztig). For any ν ∈ Λ , the retraction
M
f :X ∩f−1(XM,ν)→ XM,ν
P γ P γ γ
is an iterated affine fibration over k of relative dimension
val(det(ad(γ)|n(F))).
In particular, for P = M N ∈ P(M ), ν ∈ Λ , we have
0 0 M0
IG = |Λ\X (F )|
γ γ q
= |(X ∩f−1(XM0,ν))(F )|
γ P γ q
= qval(det(ad(γ)|n(F))) ·|(XM0,ν)(F )|
γ q
(2.3) = q21val(det(ad(γ)|gF/m0,F))·IγM0.
where the first equality is due to the fact that Λ∼= Λ , and the last equality is because γ
M0
is anisotropic in M (F).
0
2.2. Arthur’s weighted orbital integral. Now consider Arthur’s weighted orbital inte-
gral (1.1), we need to explain the weight factor v (g).
M0
2.2.1. The weight factor v . Roughly speaking, the weight factor v (g) is the volume of
M0 M0
a polytope in aG generated by the point [g] ∈X.
M0
For P = MN ∈ F(A), let H :X → aG be the composition
P M
H :X −f→P XM −ν−M→ Λ → aG.
P M M
Ithasthefollowingremarkableproperty. Thereisanotionofadjacencyamongtheparabolic
subgroups in P(M): Two parabolic subgroups P = MN , P = MN ∈ P(M) are said to
1 1 2 2
beadjacent ifbothofthemarecontainedinaparabolicsubgroupQ = LU suchthatL ⊃ M
and rk(L)= rk(M)+1. Given such an adjacent pair, we define an element β ∈ Λ in
P1,P2 M
the following way: Consider the collection of elements in Λ obtained from coroots of A
M
in n ∩n−, we define β to be the minimal element in this collection, i.e. all the other
1 2 P1,P2
elements are positive integral multiples of it. Note that β = −β , and if M = A,
P2,P1 P1,P2
then β is the unique coroot which is positive for P and negative for P .
P1,P2 1 2
Proposition 2.2 (Arthur[A]). Let P , P ∈ P(M) be two adjacent parabolic subgroups.
1 2
For any x ∈ X, we have
H (x)−H (x) = n(x,P ,P )·β ,
P1 P2 1 2 P1,P2
with n(x,P ,P ) ∈ Z .
1 2 ≥0
For any point x ∈ X, we write Ec (x) for the convex hull in aG of the H (x), P ∈
M M P
Q
P(M). For any Q ∈ F(M), we denote by Ec (x) the face of Ec (x) whose vertices
M M
are H (x), P ∈ P(M), P ⊂ Q. When M = A, we omit the subscript A to simplify the
P
notation.
To define the volume, we need to choose a Lebesgue measure on aG . We fix a W-
M0
invariantinnerproducth·,·ionthevectorspaceaG. NoticethataM andaG areorthogonal
A A M
TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS 9
to each other with respect to the inner product for any M ∈ L(A). We fix a Lebesgue
measure on aG normalised by the condition that the lattice generated by the orthonormal
M0
bases in aG has covolume 1.
M0
The weight factor v (g) is the volume of the polytope Ec ([g]) in aG . It has the
M0 M0 M0
following invariance properties: It is invariant under the right action of K, i.e.
v (gk) = v (g), ∀k ∈K.
M0 M0
This is evident from the definition of v (g). It is not so evident but also true that
M0
v (tg) = v (g), ∀t∈ T(F).
M0 M0
In fact, for any P ∈P(M ), we have
0
f (tg) = tf (g), ∀t∈ T(F).
P P
If we write t = ǫλt , with λ ∈ Λ and t ∈ T(O), then
0 M0 0
Ec (tg) = Ec (g)+λ.
M0 M0
In particular, they have the same volume.
Using the invariance properties of v (g) and the fact that dg(G(O)) = dt(T(O)) = 1,
M0
we can rewrite Arthur’s weighted orbital integral as
dg
Ad(g)−1γ v (g) = v (g),
ZT(F)\G(F)1g(O)(cid:0) (cid:1) M0 dt [g]∈ΛX\Xγ(Fq) M0
i.e. it is a weighted count of the rational points on the affine Springer fiber.
2.2.2. A variant. In their work on the weighted fundamental lemma [CL2], Laumon and
Chaudouard introduce the following variant of the weighted orbital integral. Let ξ ∈ aG
M0
be a generic element. For g ∈ G(F), they introduce the weight factor
ξ
w (g) = |{λ ∈ Λ | λ+ξ ∈ Ec ([g])}|.
M0 M0 M0
It is the number of integral points in the polytope Ec ([g])−ξ. Similar to v (g), the
M0 M0
weight factor wξ (g) is invariant under the right action of K and the left action of T(F).
M0
Consider the weighted orbital integral
dg
Jξ (γ) = Ad(g)−1γ wξ (g) .
M0 ZT(F)\G(F)1g(O) M0 dt
(cid:0) (cid:1)
Using the invariance properties of wξ (g) and the fact that dg(G(O)) = dt(T(O)) = 1, we
M0
can rewrite it as
dg
Jξ (γ) = Ad(g)−1γ wξ (g)
M0 ZT(F)\G(F)1g(O) M0 dt
(cid:0) (cid:1) dg
= dt(Λ\T(F))−1 Ad(g)−1γ wξ (g)
ZΛ\G(F)1g(O) M0 dt
(cid:0) (cid:1)
ξ
= w (g).
M0
[g]∈ΛX\Xγ(Fq)
10 ZONGBINCHEN
Because
ξ
w (g) = |{λ ∈ Λ | λ+ξ ∈ Ec ([g])}|
M0 M0 M0
= |{λ′ ∈ Λ| ξ ∈ Ec(λ′−1[g])}|,
we can further rewrite
ξ ξ
J (γ) = w (g)
M0 M0
[g]∈ΛX\Xγ(Fq)
= |{λ ∈ Λ|ξ ∈ Ec (λ−1[g])}|
M0
[g]∈ΛX\Xγ(Fq)
(2.4) = |{[g] ∈ Xν(F ) |ξ ∈ Ec ([g])}|,
γ q M0
where ν ∈ Λ is an arbitrary element.
M0
In particular, it is a plain count of a subset of X (F ). In §4.1, we will see that the
γ q
condition ξ ∈ Ec ([g]) behaves as a stability condition (We believe that it is in fact a
M0
stability condition in the sense of GIT). In particular, there is a Harde-Narasimhan type
decomposition of X associated with it.
γ
Remark 2.1. It is now the time to explain why we make the assumption that the splitting
field of T(F) is totally ramified over F. Without this assumption, the Frobenius σ ∈
Gal(k¯/k) acts non-trivially on Λ. We can only rewrite
dg
Jξ (γ) = dt(Λσ\T(F))−1 Ad(g)−1γ wξ (g)
M0 ZΛσ\G(F)1g(O) M0 dt
(cid:0) (cid:1)
ξ
= w (g).
M0
[g]∈ΛσX\Xγ(Fq)
In particular, it is no longer equal to
|{[g] ∈ Xν(F ) |ξ ∈ Ec ([g])}|.
γ q M0
Proposition 2.3 (Chaudouard-Laumon [CL2]). We have the equality
J (γ) = vol(a /X (M))·Jξ (γ).
M0 M ∗ M0
Their original statement is for the ring of adèles, but the proof carries over to local fields
word by word. We refer the reader to [CL2], §11.10–§11.14 for the proof.
3. Counting points by Arthur-Kottwitz reduction
Let Π be a sufficiently regular positive (G,M )-orthogonal family. We count the number
0
ofrationalpointsoverF ofanyconnectedcomponentXν0(Π),ν ∈Λ . Ourresultshows
q γ 0 M0
that it can be reduced to counting points on the fundamental domains FM, M ∈ L(M ),
γ 0
and the counting result depends quasi-polynomially on the truncation parameter.
