数理解析研究所講究録
第 1715 巻 2010 年 183-192

183

ON THE ARTHUR-SELBERG TRACE FORMULA
OVER FUNCTION FIELDSNGO DAC TUAN

The Arthur-Selberg trace formula for a connected reductive groupover number fields is a very powerful tool in the theory of automorphic
representations. The trace formula was first introduced by Selberg
and then it has been studied extensively in a very impressive seriesof papers of Arthur [1, 2, 3]. It has applications in the study of Lfunctions associated to certain Shimura varieties and in the study oftransfers between automorphic representations of classical groups. We
refer the reader to the excellent surveys of Arthur [4] and Labesse [12]or the very well-written book of Gelbart [9] for more details on thesubject.
It is expected by experts that a similar trace formula would holdfor function fields. When the reductive group in question is either thegeneral linear group GL $(n)$ or some inner forms of GL $(n)$ , several noninvariant versions of the trace formula over function fields have been
introduced and studied in the works of Drinfeld [7, 8], Laumon [15]and Lafforgue [13, 14]. However, it seems that the trace formula for
a general connected reductive group over function fields has not been
written in the literature. We will report our progress on the projectaiming to fill this missing part. Details will be appeared elsewhere[19, 20].
$G$

1. NOTATIONS
be a finite
Let us fix the notations. Throughout this paper, letfield of elements and $F$ be a function field over . We will denote by
with$X$ the projective smooth geometrically connected curve overthe function field $F$ . It is well known that the set of closed points
of $X$ is in bijection with the set of places of $F$ . Let be a place of$F$the completion of $F$ with, then it is non-archimedian. Denote by
the ring of
the normalized valuation,:
respect to. It is a finite extension
the residue field of
integers. Denote by2000 Mathematics Subject Classification. llF72, llF70, $22E55$ .
$F_{q}$$F_{q}$$q$$F_{q}$$x$$x$$F_{x}$$\mathcal{O}_{x}$

$x,$

$x$

$F_{x}^{\cross}arrow \mathbb{Z}$$\mathcal{O}_{x}$$\kappa_{x}$Key words and phrases. Arthur-Selberg trace formula, function fields, HarderNarasimhan filtration.

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NGO DAC TUAN

ofand the degree of the extension will be denoted by $\deg(x)$ andcalled the degree of .
We will denote by A the ring of ad\‘eles of . It is defined as the) is. An element of A (resp. of
restricted productgiven by a collection $a=(a_{x})_{x\in X}$ where for every place of $F$ , we have
, and for almost except a finite number of places of , we have). We define the degree map(resp.
, thenas follows. Let$F_{q}$$x$$F$$\prod^{J}(F_{x}, \mathcal{O}_{x})$$A^{\cross}$

$a$

$x$

$x$

$a_{x}\in F_{x}$$a_{x}\in \mathcal{O}_{x}$$F$$deg:A^{\cross}arrow \mathbb{Z}$$a_{x}\in \mathcal{O}_{x}^{\cross}$$a=(a_{x})_{x\in X}\in A^{\cross}$$\deg(a)=-\sum_{x\in X}x(a_{x})\deg(x)$.

) can be considered as a discrete subgroup of
The group $F$ (resp.
ofsits inside the subgroup). Moreover,A (resp. ofis compact.
consisting of elements of degree and the quotient
Let be a connected reductive group over $F$ . We choose a maximalofsplit torus of over $F$ and a minimal parabolic subgroupits Levi subgroupdefined over $F$ and containing . Denote bycontaining , it is defined over $F$ . By definition, a parabolic subgroup(resp. a Levi
(resp. a Levi subgroup) of is a parabolic subgroup
and a standard
subgroup $M$ ) of defined over $F$ and containing
parabolic subgroup is a parabolic subgroup of defined over $F$ andcontaining .
) be the set of roots of (resp. that of coroots of )Let (resp.
with respect to the choice of . It is a root system. Recall that we have
fixed a minimal parabolic subgroup of containing , therefore we
into two subsets: the set of positive roots
get a partition of
the set of simple roots,
Denote by
and that of negative roots
the set of corresponding simple. Denote bysays
the simple coroot associated to . There exists a
coroots and by
bijection between subsets of and standard parabolic subgroupsof . Under this bijection, the empty set will correspond to and thewhole set will correspond to .
be the split maximal
We consider a Levi subgroup $M$ of . Let
be the maximal split quotient
torus of $M$ over $F$ containing andtorus of $M$ over $F$ . The natural map $A_{M}arrow A_{M}’$ is an isogeny and(resp.
) the real vector space(resp.
we denote byadmitting $M$ as a
. For any parabolic subgroup $P$ of) to be the real vector space(resp.
Levi subgroup, we define) associated to $M$ .
(resp.
$F^{\cross}$$A^{\cross,0}$$F^{\cross}$$A^{\cross}$$0$$A^{\cross}$$F^{\cross}\backslash A^{\cross,0}$

$G$

$T$

$G$

$P_{0}$$T$$G$$M_{0}$

$T$

$P$

$G$

$G$

$M_{0}$$G$$P_{0}$$G$$G$$\Phi^{\vee}$$\Phi$$T$$P_{0}$$T$$G$$\Phi^{+}$$\Phi$$\Phi^{-}$$\triangle$$\hat{\Delta}$$\Delta=\{\alpha_{1}, \ldots, \alpha_{n}\}$$\alpha_{i}$$\alpha_{i}^{\vee}$$I$$P_{I}$$\Delta$$G$$G$$P_{0}$$\Delta$$G$$T$$a_{M}$$A_{M}’$$X.A_{M}\otimes_{\mathbb{Z}}\mathbb{R}$$a_{M}^{*}$$G$$X^{*}A_{M}\otimes_{Z}\mathbb{R})$$a_{P}$$\alpha_{M}^{*}$$A_{M}$$a_{P}^{*}$$a_{M}$

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TRACE FORMULA OVER FUNCTION FIELDS

Let and be two parabolic subgroups of such that $P\subset Q$ . The
natural maps $A_{M_{P}}arrow A_{M_{Q}}arrow A_{M_{Q}}’arrow A_{M_{P}}’$ induces an injection
and a surjection
such that the composition
is identity. Hence we obtain a decomposition. Similarly, we have a canonical decompositionIn
particular, if we take $P=P_{0}$ and $Q=G$ , we get vector spaces
and. The set
(resp.
) forms a basis of(resp.
). The dual
basis ofwill be denoted byand contain fundamental weightsof , says. Similarly, we get the basis of fundamental
coweights of.
Now, we consider a standard parabohc subgroup of , then
. Denote by
the subset of corresponding to . The previous
discussion gives a canonical decomposition. The setof simple roots forms a basis of the real vector space. By abuseof notations, we still denote by
the image of
via the projection, it forms also a basis of. Let
be an element inwhich is the image of a simple root
under the above projection.
We will denote by
the projection ofinand call it the simple
coroot of. We will denote by) the set of(resp.
projections of fundamental weights(resp. fundamental coweights
such that. They form another basis ofand .
Let $P$ and be two parabolic subgroups of such that $P\subset Q$ andthe subsets of corresponding to and . Since $P\subset Q$ , we
. We will denote byhave(resp.
) the image under thecanonical projectionof simple roots(resp. of fundamental weights). They forms
withdifferent basis for the vector space
Similarly, we can define twodifferent basis
andof .
Following Langlands and Arthur, we can introduce the acute chamber
and the obtuse chambe
as follows:
of
$P$

$Q$

$G$

$a_{Q^{c}}arrow a_{P}$$a_{P}arrow\alpha_{Q}$$\alpha_{Q}\mapsto$$\alpha_{P}arrow\alpha_{Q}$$\alpha_{P}=a_{Q}\oplus$$\alpha_{P}^{Q}$$\alpha_{P}^{*}=a_{Q}^{*}\oplus a_{P}^{Q*}$$a_{P_{0}}^{G*}$$\Delta^{\vee}$$\triangle$$a_{P_{0}}^{G}$$\alpha_{P_{0}}^{G*}$$G$$a_{P_{0}}^{G}$$\triangle\wedge$$\triangle^{v}$$\{\varpi_{1}, \ldots, \varpi_{n}\}$$a_{P_{0}}^{G}$

$P$

$P$

$\triangle_{P}$$G$$P_{0}\subset$$P$$\triangle$$a_{P_{0}}^{G*}=\alpha_{P}^{G*}\oplus\alpha_{P_{0}}^{P*}$$\Delta$$\alpha_{P_{0}}^{G*}$$\triangle_{P}$$a_{P_{0}}^{G*}arrow a_{P}^{G*}$$\triangle_{P}$$\alpha_{P}^{G*}$$\triangle_{P}$$\alpha$$\beta\in\triangle$$\alpha^{\vee}$$a_{P}^{G}$$\beta^{\vee}$$\wedge$$\triangle_{P}\wedge$$\alpha\in\triangle_{P}$$\triangle_{P}^{v}$$\varpi_{i}$$\alpha_{i}\in\triangle_{P}$$\varpi_{i}^{\vee})$$a_{P}^{G*}$

$G$

$Q$

$\triangle_{P},$$a_{P}^{G}$$P$$\triangle$$\triangle_{Q}$$Q$$\wedge$$\triangle_{P}^{Q}$$\triangle_{Q}\subset\triangle_{P}$$\triangle_{P}^{Q}$$\alpha_{P_{0}}^{G*}arrow a_{P}^{G*}arrow a_{P}^{Q*}$$\alpha\in\triangle_{P}-\triangle_{Q}$$\alpha\in\Delta_{P}-\triangle_{Q}$$\varpi_{\alpha}$$\alpha_{P}^{Q*}$$\triangle_{P}^{Q\vee}$$\triangle_{P}^{Q\vee}\wedge$$\alpha_{P}^{Q}$$r^{+}a_{P}^{Q}$$\alpha_{P}^{Q+}$$\alpha_{P}^{Q+}=\{p\in\alpha_{P}^{Q+}:$$+_{a_{P}^{Q}}=\{p\in\alpha_{P}^{Q+}:$$\alpha_{P}^{Q}$$\langle\alpha,p\rangle\geq 0$$\langle\varpi,p\rangle\geq 0$

for all

for all

Remark that we always have the inclusion$\alpha\in\Delta_{P}^{Q}\}$$\varpi\in\triangle_{P}^{Q}\}\wedge$$a_{P}^{Q+}\in^{+}a_{P}^{Q}$.

2. HARDER-NARASIMHAN FILTRATION
We choose a maximal compact subgroup $K$ of $G(A)$ as in [3, section: $M(A)arrow$
1 . Let $M$ be a Levi subgroup of . We define the map
as follows. Let $m$ be an element of $M(A)$ . To each character$]$$\alpha_{M}$$G$$H_{M}$

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NGO DAC TUAN

: $Marrow G_{m}$ of $M$ defined over $F$ , we get an integer $\deg(\chi(m))$ .
Hence, we obtain an element $H_{M}(m)\in a_{M}$ such that$\chi$$\langle H_{M}(m),$ $\chi\rangle=\deg\chi(m)$.

Remark that the homomorphism
takes value in a lattice of the real.
vector space
Next, let be a parabolic subgroup of , with Levi subgroup $M$ andradical unipotent subgroup $N$ . We extend the previous homomorphism: $G(A)arrow\alpha_{M}$ . Since $G(A)=P_{0}(A)K$ , we also
to get a map
have $G(A)=P_{0}(A)K$ . Hence any element $g\in G(A)$ can be written$g=nmk$ with $m\in M(A),$ $n\in N(A)$ and $k\in K$ , and we define$H_{P}(g)=H_{M}(m)$ .
We review the notion of orthogonal family appeared in the work of
Arthur. Let $M$ be a Levi subgroup of . We consider the real vector
space
defined in the first section. The rootsof $M$ definesand divides
different hyperplanes in
into chambers. The open
thechambers are called Weyl chambers. We will denote by
set of parabolic subgroups of admitting $M$ as a Levi subgroup. It
is in bijection with the set of Weyl chambers,
is known that
therefore we say that two elements and $P’$ inare adjacent if
the corresponding Weyl chambers are adjacent. Furthermore, there isa unique root
such that is a simple root for $Pand-\alpha$ is a
simple root for $P’$ .
. We say that thisbe a family of points inLetfamily is $(G, M)$ -orthogonal if the following condition is satisfied: forevery pair of adjacent parabolic subgroups
and $P’$ , we denote by
the unique simple root of $P$ such thatis a simple root of $P’$ ,.
then we require thatfor some real number
If we require further that $m_{P}\geq 0$ , then the family is said positive$(G, M)$ -orthogonal.
We provide some examples to illustrate the previous definition. In(resp. in the Weyl chamber
the first example, let be a point in. We take the family of Weyl orbit of . Then it is $(G, M)-$
orthogonal (resp. positive $(G,$ $M)$ -orthogonal). In the second example,
we consider an element $g\in G(A)$ . Then the family
is positive $(G, M)$ -orthogonal.
We will recall now a result of Behrend on positive $(G, M_{0})$ -orthogonal
be a positive $(G, M_{0})$ -orthogonal family.
families. Let. We
Let
be a parabolic subgroup of , not necessarily in$H_{M}$$\alpha_{M}$

$P$

$G$

$H_{P}$$G$$\Phi_{M}$$\alpha_{M}^{G}$$a_{M}^{G}$$a_{M}^{G}$$\mathcal{F}(M)$$G$$\mathcal{F}(M)$$P$$\alpha\in\Phi_{M}$$\mathcal{F}(M)$$\alpha$$a_{M}^{G}$$\lambda=\{\lambda_{P}\}_{P\in F(M)}$$P$$-\alpha$$\alpha$$\lambda_{P}-\lambda_{P’}=m_{P}\alpha^{\vee}$$m_{P}\in \mathbb{R}$$\underline{\lambda}$$a_{M}^{G}$$\lambda$$a_{M}^{G+})$$\lambda$$\underline{\lambda}$$\underline{\lambda}=\{H_{P}(g)\}_{P\in F(M)}$$\underline{\lambda}=\{\lambda_{P}\}_{P\in \mathcal{F}(M_{0})}$

$P$

$G$

$\mathcal{F}(M_{0})$187
TRACE FORMULA OVER FUNCTION FIELDS

define$R(P)=\{\beta\in\Phi$:$[ \beta]_{P}=\sum_{\alpha\in\Delta_{P}}m_{\alpha}\alpha$with$m_{\alpha}\geq 0\}$and$\lambda_{P}\rangle$$\deg_{P}(\underline{\lambda})=\sum_{\beta\in R(P)}\langle\beta,$.

With these notations, Behrend proved [5]:Theorem 2.1 (Behrend). There exists a unique pambolic subgroupof verifying the following conditions:a For every pambolic subgroup $QofG$ such that $Q\in P$ andcontains a unique simple root, says , then we require

$P$

$G$

$)$

$\triangle_{Q}-\triangle_{P}$$\alpha$$\langle\alpha,$$\lambda_{P}\rangle\leq 0$.

b The element
lies in the Weyl chamber.
Further, it is chamcterized by the following property: we consider
the set of pambolic subgroupsof such that$)$$a_{P}^{G+}$$\lambda_{P}$

$P$

$G$

$\deg_{P}(\underline{\lambda})=\max\{\deg_{Q}(\underline{\lambda})\}Q$

where we consider all pambolic subgroups
maximal element of this set.

$Q$

of

$G$

, then $P$ is the unique

We will apply Behrend theorem to our situation. Our goal is to define the notion of Harder-Narasimhan polygon associated to an element$g\in G(A)$ . It could be considered as a generalization of the classical
Harder-Narasimhan polygon associated to vector bundles over curves.
We have to introduce some more notations. Recall that for every
element of the relative Weyl group $W$ of , we have fixed a representative $w_{s}\in G(F)$ of . By the choice of the maximal open compact
subgroup $K$ , Arthur proved [3, Lemma I.I] that there exists a uniquepoint
such that, for every element
considered as a point in$s\in W$ , we have
$s$

$G$

$s$

$s$

$T_{0}\in\alpha_{M_{0}}^{G}$$\alpha_{M_{0}}$$H_{\Lambda l_{0}}(w_{s}^{-1})=T_{0}-s^{-1}T_{0}$.

We remark that if
is split over , then we can choose $K=$
. It implies that the elements
can be chosen in $K$ andhence $H_{M_{0}}(w_{S}^{-1})=0$ . Consequently, $T_{0}=0$ . However, for general.
connected reductive group over $F,$
be a parabolic subgroup
Now, let be an element in $G(A)$ and. There exists
but not necessarily containingof defined over$G$

$F_{q}$$\prod_{x}G(\mathcal{O}_{x})$$w_{s}$

$G$

$Q$

$g$

$G$

$T_{0}\neq 0$$F$$M_{0}$

188
NGO DAC TUAN$\delta\in G(F)$and$P\in \mathcal{P}(M_{0})$such that$P\subseteq Q’$. We define:$:=\delta Q\delta^{-1}$$\deg_{Q}(g):=(\sum_{\alpha\in R(Q’)} or, [H_{P}(\delta g)]_{P}^{G}-T_{0})$.

implies that the degree $\deg_{Q}(g)$ does not
The previous choice of.
depend on our choicedeconsisting of parabolic subgroups
We introduce the set
such that $\deg_{P}(g)=$defined over $F$ , but not necessarily in$\max_{Q}\deg_{Q}(g)$ . It follows from Behrend
previous result that this set
contains a unique maximal element. Remark that the set of parabolic
subgroups of defined over $F$ is in bijection with the set of pairs
where is a standard parabolic subgroup of and is a class in thequotient $P(F)\backslash G(F)$ . Under this identffication, we can reformulate
be a pair as before. We define
our previous result as follows. Let
the polygon$T_{0}$$(\delta, P)$$P$$\mathcal{A}$$G$$\mathcal{P}(M_{0})$

$s$

$G$

$(P, \delta)$

$G$

$P$

$\delta$$(P, \delta)$$p_{P}^{\delta g}=[H_{P}(\delta g)]_{P}^{G}-[T_{0}]_{P}^{G}$

It is a point in$a_{P}^{G}$.

. We have proved:

Theorem 2.2. We keep the previous notations. There exists a uniquecalled the canonical Harder-Narasimhan filtmtion associatedpair
to $g\in G(A)$ which satisfies the following conditions:andsuch thata For every standard pambolic subgroup,contains a unique simple mot and for every
we have the inequality.
$(P, \delta)$$Q\subseteq P$

$Q$

$)$

$\Delta_{Q}-\triangle_{P}$$\delta_{Q}\in Q(F)\backslash P(F)$$\alpha_{f}$$\langle\alpha,p_{P}^{\delta_{Q}\delta g}\rangle\leq 0$

b The element
The element$)$$p_{P}^{\delta g}$.
lies in the Weyl chamberconsidered as a point in$a_{P}^{G+}$$p_{P}^{\delta g}\in a_{P}^{G}$$a_{M_{0}}^{G}$is called theHarder-Namsimhan polygon of $g\in G(A)$ .

3. $HARDER’ S$ REDUCTION THEORY AND TRUNCATION PARAMETERS. Let $g\in G(A)$ andLet be a point in the acute chamberandbe a pair where $P$ is a standard parabolic subgroup ofclass in the quotient $P(F)\backslash G(F)$ . We have defined$a_{M_{0}}^{G+}$

$p$

$G$

$p_{P}^{\delta g}=[H_{P}(\delta g)]_{P}^{G}-[T_{0}]_{P}^{G}$

We say that$p_{P}^{\delta g}>Pp$

if we have$(P, \delta)$$\delta$is a.

$p_{P}^{\delta g}-\lceil p]_{P}^{G}\in+a_{P}^{G}$.

Definition 3.1. We keep the previous notations. Suppose further thatis the canonical Harder-Narasimhan filtration associated to andis the associatedconsidered as a point inthat $p(g)$$(P, \delta)$$g$$:=p_{P}^{\delta g}\in a_{P}^{G}$$\alpha_{M_{0}}^{G}$

189
TRACE FORMULA OVER FUNCTION FIELDSHarder-Narasimhan polygon. We will say that $p(g)$ is bounded by ,write $p(g)\leq p$ if $p-p(g)\in+a_{M_{0}}^{G}$ .
$p$

We review the reduction theory for function fields entirely due toHarder. For the references, we send the reader to [16, section 4] foran excellent overview and to the original papers of Harder [10, 11] for
proofs. For a real number , we consider the following set$c$$P_{0}(c)=\{g\in P_{0}(A):(\alpha, H_{M_{0}}(g))\geq c, \alpha\in\triangle\}$.

Harder proved two fundamental and deep results of the reduction theory over function fields:

Theorem 3.2 (Harder). There exists a constant $c>-\infty$ such that
$G(A)=G(F)P_{0}(c)K$ . The set
such that we have the previous
property is called a Siegel domain.
$P_{0}(c)$Obviously, if
is a Siegel domain for some real number , thenwill be a Siegel domain for any real number with $d\leq c$ .
$P_{0}(c)$

$c$

$c’$

$P_{0}(c’)$

Theorem 3.3 (Harder). Let be a real number such that $G(A)=$$G(F)P_{0}(c)K$ . Then there exists a positive constant $c_{2}=c_{2}(c)$ depending on verifying the following property: If $\gamma\in G(F)$ and $x\in P_{0}(c)$, then $\gamma\in P(F)$ where the pambolic subgroup $P$
such thatconsisting of all simple roots withcorresponds to the subset I of
.
$c$

$c$

$\gamma x\in P_{0}(c)$$\Delta$$\alpha$$(\alpha, x)\geq c_{2}$Harder’s reduction theory imphes immediately that the subset of$G(F)\backslash G(A)/J$ consisting of elements
such that $p(g)\leq p$ is compact.
$g$

4. THE NON-INVARIANT TRACE FORMULA: RESULTS ANDDISCUSSION

We consider the function space $L^{2}(G(F)\backslash G(A))$ which consists ofsquare integrable complex functions on $G(F)\backslash G(A)$ . Let $f\in C_{c}^{\infty}(G(A))$
be a function over $G(A)$ which is locally constant and of compact support. We choose a Haar measure on $G(A)$ and define an operator $R(f)$on $L^{2}(G(F)\backslash G(A))$ as follows:$(R(f) \varphi)(x)=\int_{G(A)}f(y)\varphi(xy)dy$.

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NGO DAC TUAN

A simple calculation implies that$(R(f) \varphi)(x)=\int_{G(A)}f(y)\varphi(xy)dy$$\sum f(x^{-1}\gamma y)\varphi(y)dy$$= \int_{G(F)\backslash G(A)_{\gamma\in G(F)}}$.

Hence, this integral operator $R(f)$ admits a kernel $K(x, y)$ given by
the formula:.
$K(x, y)= \sum_{\gamma\in G(F)}f(x^{-1}\gamma y)$

Since is locally constant and of compact support, the sum vanishes
for all except a finite number of $\gamma\in G(F)$ , therefore the kernel is welldefined.
We consider the kernel over the diagonal and define$f$$k(x)=K(x, x)= \sum_{\gamma\in G(F)}f(x^{-1}\gamma x)$

as the function over

$G(F)\backslash G(A)$. Remark that this function is invariant

by multiplication with $Z(A)$ where is the center of . We will introduce a discrete torsion-free subgroup of $Z(A)$ such that $JZ(F)\backslash Z(A)$is compact and $Z(F)\cap J=0$ . In general, the integral
diverges and the goal of the trace formula and its refinements is toregularize this integral.
We follow closely Arthur’s procedure to define the modified kernel. Let be a standard parabolic subgroup of . Following Arthur,
we consider the integral operator $R_{P}(f)$ acting on the function space$L^{2}(M(F)N(A)\backslash G(A))$ defined as follows:
$Z$

$G$

$J$

$\int_{JG(F)\backslash G(A)}k(x)$

$P$

$G$

$(R_{P}(f) \varphi)(x)=\int_{G(A)}f(y)\varphi(xy)dy$.

It has a kernel given by the formula$f(x^{-1}\gamma ny)dn$$\sum$$K_{P}(x, y)= \int_{N(A)_{\gamma\in M(F)}}$for every $y\in M(F)N(A)\backslash G(A)$ . Denote by $k_{P}(x)=K_{P}(x, x)$ , inother words, for every $x\in M(F)N(A)\backslash G(A)$ ,$x,$$k_{P}(x, y)= \int_{N(A)}\sum_{\gamma\in M(F)}f(x^{-1}\gamma nx)dn$.

. Suppose that it is
Let be a point in the acute chamber
is sufficiently large with
sufficiently regular in the sense that$a_{M_{0}}^{G+}$$p$$\langle\alpha,p\rangle$

191
TRACE FORMULA OVER FUNCTION FIELDS

respect to the support of . We define the Arthur’s modified kernel$f$$k^{p}(g)= \sum_{P_{0}\subseteq P}(-1)^{\dim a_{P}^{G}}\sum_{\delta\in P(F)\backslash G(F)}1(p_{P}^{\delta g}>Pp)k_{P}(\delta g)$.

It is a function over $G(A)$ which is invariant on the left by the subgroup$JG(F)$ . Hence, we will consider the modified kernel as a function overthe double quotient $G(F)\backslash G(A)/J$ .
In contrast to the number field setting, we will show that, in thefunction field case, the modified kernel over $G(F)\backslash G(A)/J$ is in fact the
product of the original kernel and the characteristic function of a compact set. Consequently, it is automatically integrable over $G(F)\backslash G(A)/J$ .

Theorem 4.1. We keep the previous notations. Then for every $g\in$$G(A)$ , we have the following equality:$k^{p}(g)=1(p(g)\leq p)k(g)$ .

The modified integral will be defined as follows:$J^{p}(f)=l_{G(F)\backslash G(A)}k^{p}(x)$.

We have proved that our modified integral converges. The modified
kernel has an obvious geometric expression. It also has a spectral expression following from deep results of Langlands and Morris [16, 17, 18]called the Langlands decomposition theorem. Hence, we get the firststep of our desired non-invariant trace formula, which states an equality
between a geometric side and a spectral side.
Next, we need to refine the previous trace formula. On the geometricside, Arthur proposed to decompose the kernel into semisimple conjugacy classes. However, since we are working with the function field
which is not perfect, we cannot apply directly Arthur’s procedureand it would be an interesting question how to overcome this difficulty.
Let us attract our attention to the spectral side. It turns out that afine spectral refinement could be obtained using only the Langlands
decomposition theorem. It has been proved by Lafforgue for GL $(n)$[15, 13] and then generalized by the author for reductive groups over$F[19,20]$ .
Finally, it would be very interesting to investigate the behavior of
in function of the truncating parameter for fixed . In thenumber field setting, the parameter takes value in some open chamberof a real vector space and Arthur proved that the function
is infact a polynomial of . In our function field setting, the parameter$F$$J^{p}(f)$

$p$

$f$

$J^{p}(f)$

$p$

$p$

192
NGO DAC TUAN

takes value in a lattice of a real vector space, but we can still provethat theis a quasi-polynomial of $p[20]$ .
$J^{p}(f)$

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[18] C. MOEGLIN, J.-L. WALDSPURGER, D\’ecomposition spectrale et s\’eriesd ‘Eisenstein. Une paraphmse de l’Ecriture, Progress in Mathematics, 113(1994), Birkhauser Verlag, Basel.
[19] T. NGO DAC, Sur le developpement spectral de la formule des tracesd’Arthur-Selberg sur les corps de fonctions, to appear in Bull. Soc. Math.
Rance, available at www.math.univ-paris13.fr $/\sim ngodac/traceformula.pdf$ .
[20] T. NGO DAC, On the Arthur-Selberg trace formula over function fields, in
preparation.

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