The Trace Formula and its Applications
An introduction to the work of James Arthur∗
I balanced all, brought all to mind,
An Irish airman foresees his deathW. B. Yeats

In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded
the Canada Gold Medal by the National Science and Engineering Research Council. This is a highhonour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to amathematician, and the choice of Arthur, although certainly a recognition of his great merits, is also a
recognition of the vigour of contemporary Canadian mathematics.
Although Arthur’s name is familiar to most mathematicians, especially to those with Canadianconnections, the nature of his contributions undoubtedly remains obscure to many, for they are all tiedin one way or another to the grand project of developing the trace formula as an effective tool for the
study, both analytic and arithmetic, of automorphic forms. From the time of its introduction by Selberg,
the importance of the trace formula was generally accepted, but it was perhaps not until Wiles used
results obtained with the aid of the trace formula in the demonstration of Fermat’s theorem that it had
any claims on the attention of mathematicians – or even number theorists – at large. Now it does, andnow most mathematicians are willing to accept it in the context in which it has been developed by
Arthur, namely as part of representation theory.
The relation between representation theory and automorphic forms, or more generally the theory of
numbers, has been largely misunderstood by number theorists, who are often of a strongly conservative
bent. Representation theory appears, in number theory and elsewhere, because symmetries create
redundance, and this disorderly redundance has to be reduced to an orderly, perceptible uniqueness.
Physicists and chemists, who are often bolder than mathematicians, accepted this and incorporated
unhesitatingly representation theory into their analysis. In spite of Gauss, Dirichlet, Dedekind andFrobenius, number theorists balked and sought alternatives, but in the hope that such obscurantism
belongs to the past, I am going to begin immediately with representations, and with the language ofadèles, adding only – as a corrective – that there are circumstances, even with the trace formula, when it∗ During the preparation of this introduction I was able to proﬁt from frequent conversations with Arthur himself.
I am grateful for the patience with which he responded to my many questions.
Appeared in Canad. Math. Bull., vol. XX (2001).

Introduction to the work of James Arthur

2

is useful to calculate in as concrete a manner as possible, and that then, in spite of the great conceptualease they offer, representations and adèles have to be replaced by bases and by congruences. But thesecircumstances are not so frequent as many would like and do not arise at all in the work of Arthur.
Arthur, like Harish-Chandra, is inclined to work not with particular groups but with a general
reductive group. The generality sometimes veils the difficult technical or conceptual achievements that
inhere in various of his results. Nevertheless, any attempt to present an outline of his theory in termsof specific groups would become entangled in a welter of largely irrelevant detail, so that I have chosen
to assume that the reader is familiar, at least at a formal level, with the basic concepts from the theoryof algebraic groups and from other pertinent domains.
Although I have tried here to expound Arthur’s papers in a coherent, logical sequence and to
avoid all fatuous metaphor, I expect – and encourage – most readers to skim the report. Some passages,
unfortunately far too many, are laborious, and of interest only to those who are contemplating a serious
study of the domain, but others can, I hope, be read quickly and easily and will give even those with notime and no inclination to trouble themselves with details some notion of Arthur’s accomplishments.1
Selberg first introduced the trace formula to study the asymptotic properties of the spectrum of
automorphic forms for the group SL(2), and then went on to apply it to other analytic problems, inparticular to the eponymic zeta-function and to some automorphic L-functions, but Arthur has beenmuch more concerned with questions associated with functoriality, thus the comparison of automorphic
forms or representations for different groups. Although the trace formula for groups with compact
quotient is the result of a simple formal analysis, the formula for groups with noncompact quotient, andthus with continuous spectrum, is fraught with difficult analytic problems, some appearing alreadyfor groups of rank one, in particular for SL(2), others, of a different nature, appearing first for groupsof higher rank. Although not the very first, one of the earliest signs of Arthur’s mathematical powers
was his derivation in [5], [7], [9] of an identity, valid for a general reductive group, that was a genuine
trace formula, thus an identity with a sum over conjugacy classes on one side, and a sum over spectral
terms on the other. These two sides are usually referred to as the geometric and the spectral sides of
the trace formula.

1. The first general trace formula. Recall that the theory of automorphic forms, in its purely analytic
formulation, refers to the study of invariant functions on Γ\G, where for the present purposes – with
1 It may be manifest from this report, but I nonetheless warn the innocent reader that my own knowledge andunderstanding of Arthur’s papers is far less than it should be. Much of the following that is obscure would otherwise
be clear. The trace formula cries out for competent exposition, as a whole or in part, and for smoother, more direct
treatments of a number of issues that Arthur overcame only by circuitous means.

Introduction to the work of James Arthur3

a notation that will trouble the purist – Γ = G(Q) and G = G(A), the group G on the right being aconnected reductive algebraic group over Q. (Unless the contrary is explicitly indicated, the symbol

G standing alone without argument will denote, at least in the first part of this report, the topologicalgroup G(A).) One could replace the field Q by any finite extension F of it, thus by an algebraic number
field, but because of the possibility of reducing the scalar field back to Q, this is not a more generalsituation. On the other hand, F could also be a function field, a possibility not covered by Arthur’stheory and not yet, so far as I know, treated.
The structure of G as a topological group is, like that of A itself, far from simple, but rather than
discuss it, I simply take various attendant notions as self-evident. If f is a (smooth) function (withcompact support) on G it defines an operator


φ→φ,





φ (x) =

φ(xy)f (y)dhon the functions on Γ\G with kernel(1)

K(x, y) =f (x−1 γy)γ∈Γ

We can think of this as an operator on L2 (Γ\G). It will not be of trace class unless Γ\G is compact,
but that is seldom the case. There are two sources of noncompactness. The centre of the algebraic
group G may contain a split torus or there may be a split torus in its derived group. If it were not thatthe principal applications of the theory were to groups with centre, it would be best to avoid the firstpossibility simply by assuming that the algebraic group is semisimple. I do this for now, observing that
otherwise we can replace G by a subgroup defined by some conditions modulo the derived group, or
we can form, rather than L2 (Γ\G), an L2 -space of functions transforming according to a character ofa subgroup of the centre, or we can employ, as Arthur does to gain flexibility, some combination of thetwo devices.
The existence of a split torus in the derived group is a more serious matter and entails a truncationof the kernel K that is dictated by the general reduction theory developed by Siegel and completed
by Borel and Harish-Chandra and that (for groups of rank greater than one) is one of the key earlier
inventions of Arthur. The central result of reduction theory is that grosso modo the geometry of Γ\Gis that of a cone in the Lie algebra a over R of a maximal Q-split torus A of the semisimple part of thealgebraic group G. This cone is defined by the set ∆ of simple roots of the torus, thus as(2){H | α(H) > 0,α ∈ ∆}.

Introduction to the work of James Arthur4

ˆ = {α } of dual roots defined by
In addition to the set of roots, there is also the set ∆α(β ) = δα,β .
It is common in the theory to fix a minimal parabolic subgroup over Q that contains A and to
consider only parabolic subgroups P that are defined over Q and that contain this fixed minimal one.
The subsets of ∆ are then identified with these parabolic subgroups. So we denote them by ∆P . Since

P contains A, it has a distinguished Levi factor M . Let N be its unipotent radical.
It is usual to distinguish a maximal compact subgroup K of G. Then any element of G can be
written as x = nmk with n in the unipotent radical of a given P , m in its Levi factor and k in K . The
vector space a can be identified with the set of real-valued functions on ∆ and those functions thatvanish on ∆P form then the Lie algebra aP over R of the centre of M and, in addition, can be thoughtof as the space of homomorphisms of the group of characters of M defined over Q into R. Thus if mlies in M , we can define HM (m) = H(m) in aP byeH(m),χ = |χ(m)|.
The mapx → HP (x) = HM (m)allows us to pull functions on aP back to functions on G.
If the trace of the operator defined by f or the kernel K were defined, it would be the integralof K(x, x) over a fundamental domain. In general it is not, because the spectrum has continuouscomponents in all dimensions up to dim(A). An analysis of the asymptotic behaviour of K(x, y) ona fundamental domain, thus essentially on a cone as in (2), suggests a truncation of K(x, y) that isdefined with the help of the simple roots and their duals and the algebraic groups P . Both the kernel
and its truncation can be expressed in two different ways, as a sum over classes in Γ that remain to be
introduced or as a spectral sum. Integrating one or the other expression of the truncated kernel overthe diagonal yields two results that must be equal. This equality or identity is the basis for all further
development. So we begin by reviewing the two expressions it, the geometric side given by a sum overclasses in Γ and the spectral side.
Two elements in Γ are said to be in the same class o if their semisimple components are conjugate
in Γ. Thus for example all unipotent elements have as semisimple component the unit element andIntroduction to the work of James Arthur

5

therefore form a single class. The truncation is defined by an element T deep in the cone (2), thus withα(T )  0 for all α ∈ ∆. If P is a parabolic subgroup we set
KP (x, y) =

 
M (Q)

N (A)

f (x−1 γny)dn.

If ∆ contains a single element, thus if the dimension of A is one so that the fundamental domain isessentially a half-line, then the truncation is straightforward. We take P to be the minimal parabolicand subtract from the original kernel, or rather its restriction to the diagonal, the expression

(3)

KP (δx, δx)τ̂P (H(δx) − T ),P (Q)\G(Q)in which τ̂P is the characteristic function of a half-line. In general, such expressions are formed for each

P and alternately added and subtracted.
If o is a class, we can introduce the partial sum

Ko (x, y) =f (x−1 γy)γ∈oas well as the truncations of the partial sums. Thus the truncation itself kT (x, f ), a function on the
diagonal and thus of x alone, becomes a sum over the set O of classes o,k T (x, f ) =koT (x, f ).

o

The integration over x can proceed term by term and yields the first form of the geometric side of the
trace formula.
Thus the basic objects on the geometric side, at least in its first form, are the integrals


JoT (f ) =

(4)

Γ\G

koT (x, f )dx.

Expressing these integrals in a usable form was one of the main tasks undertaken by Arthur in theyears following the appearance of the first papers. In some cases, however, a suggestive form lies closeat hand.
Suppose that o is an equivalence class that consists entirely of semisimple elements. We can chooseγ ∈ o and a P with Levi factor M such that γ is contained in no proper parabolic subgroup of M over

Introduction to the work of James Arthur

6

Q. If the centralizer of γ is contained in M the class is called unramified. Then, apart from a factor
given by a volume, the expression (4) is equal to a weighted orbital integral,(5)Gγ (A)\G(A)f (x−1 γx)v(x, T )dx.

The weights v(x, T ) can be made explicit, and we shall return to this in the course of our discussion,
which will be lengthy because there are a large number of defects in the geometric side as it now stands
and in (5). First of all, we need explicit expressions for (4) for all classes, not just the unramified ones.
Secondly, even for the unramified ones, the term (5) is not yet amenable to local harmonic analysis
and almost all applications of the trace formula reduce ultimately to local comparisons. Moreover,the parameter T has been introduced simply to effect a truncation and has no intrinsic significance,
so that it should not appear in a final result. In addition, in order to truncate, we have introduced afixed maximal compact subgroup of G, so that neither the geometric nor the spectral side of the trace
formula as it at first appears is invariant. Thus they certainly cannot be traces. Since it is traces that are
to be compared, we have to see about passing from the noninvariant form to an invariant form. Before
undertaking all these modifications, we examine the spectral side.
The spectral side, as its name implies, entails a knowledge of the spectral decomposition ofL2 (Γ\G), in other words of its decomposition into a direct integral of irreducible representations of
G. The basic structure of the space is defined in terms of the parabolic subgroups P . In particular, thespace of cusp forms is defined by the vanishing of the integrals over their unipotent radicals,φ(ng)dn.
N (Q)\N (A)The space of cusp forms is invariant and irreducible under the action of G and the representation
on it is a discrete direct sum of irreducible representations. Many questions can be asked about thecomponents, but that is an arithmetical problem not part of the spectral theory. The spectral theory
takes the structure of the space of cusp forms for G and for the Levi factors M = MP of the parabolic
subgroups P as given and constructs in terms of these unknown objects, whose deeper properties it
is now the goal of the trace formula to investigate, the full decomposition of L2 (Γ\G). Although weassumed, for simplicity, that G was semisimple, this will no longer be so for the groups M . The centreof M will contain a split torus AP whose dimension is equal to that of aP . As a result, if ρ is an
automorphic representation of M and λ an element in the dual of aP thenρλ = ρ ⊗ eiλ(H(m))

Introduction to the work of James Arthur

7

is also an automorphic representation of M . According to the theory of Eisenstein series, the representations induced from the ρλ , ρ cuspidal, all appear in the decomposition of L2 (Γ\G), so that there isa continuous spectrum of dimension dim aP associated to the pair (M, ρ). There is a simple notion ofequivalence on the pairs (M, ρ) with ρ cuspidal. An equivalence class will be denoted χ and the set of
all classes by X. To each such class is associated an invariant subspace L2χ (Γ\G) of L2 (Γ\G) andL2 (Γ\G) = ⊕L2χ (Γ\G).
The space L2χ (Γ\G) may contain not only continuous spectrum of dimension dim aP but possibly
spectra of any dimension between 0 and dim aP . The exact nature of this spectrum is unknown, butthere is a conjecture of Arthur that was suggested by his reflections on the trace formula and that is
related to Ramanujan’s conjecture, or rather to its failure for general groups in the first, crude form inwhich it was proposed. This conjecture, to which we shall return, allows a calculation of the precise
spectrum of the space L2χ (Γ\G) from a knowledge of certain automorphic L-functions. Some cases
have been established, so that there is place for confidence in it, but a general proof will probably
demand a much deeper understanding of trace formulas than we have at present. The projections on
the spaces L2χ (Γ\G) entail a decomposition of the kernel K(x, y) asK(x, y) =Kχ (x, y),χ∈Xand the truncation can be performed on the individual Kχ , so thatk T (x, f ) =kχT (x, f ).

Arthur establishes, but his proof is not direct, that this decomposition can be integrated term byterm, so that(6)

JoT (f ) =o∈OJχT (f ),χ∈X

where O is the collection of classes o and
(7)JχT (f ) =Γ\GkχT (x, f )dx.

The formula (6) is the first form of the trace formula, but is useless without a better understanding of
the terms (4) and (7).

Introduction to the work of James Arthur

8

Just as it will be useful and instructive to have the evaluation (5) of (4) in the simplest case, so will
it be useful to have an evaluation of (7) in the simplest case, for χ unramified – in a sense that need not
be defined more precisely here. Then the space L2χ (Γ\G) does not contain the additional spectrum of
dimension less than dim aP . It may, and usually does, happen that there are several P and ρ such that(M, ρ) is in the class χ. ThenJχT (f ) =

(8)


P

1
n(AP )Π(M )tr(ΩTP (π)χ · IP (π, f )χ )dπ.

The formula is from [9] but the notation is that of the paper [13]. Although I have modified it a little in
an attempt to make it more transparent, it remains less than ideal. The sum is over the pertinent P andn(AP ) is an integer, the number of chambers in aP . The integration appears at first glance to be overthe unitary dual of M (A). If IP (π) is the representation of G unitarily induced from π thenIP (π, f ) =f (x)IP (π, x)dx.
G

The expression IP (π, f )χ is defined similarly in terms of IP (π)χ , which is 0 if (P, π) does not lie in χ
and is IP (π) with an appropriate multiplicity if it does.
The central objects in (8) are the operators ΩTP (π)χ . They are in essence logarithmic derivativesof intertwining operators. Because of the presence of a derivative, the expression (8) is not invariant.
Thus it has the same defects as (5). Not only does it depend on the ancillary and ultimately irrelevant
parameter T but also it is not invariant.
Since Arthur’s bibliography is long, it may appear that we are spending far too much time with
only two of the very earliest papers and that we are never going to get off the ground. Nonetheless, if
we are to understand at all the nature of the objects with which Arthur had to deal, we have to look at(5) and (8) more closely.

2. The geometry of convex sets defined by roots and coroots. The sum in (6) – of either theleft or the right side – turns out to be a polynomial in the variable T . The term of principal interest in
this polynomial is the constant term. The higher-order terms in T can be interpreted as the constantterm of a trace formula for functions on the Levi factors that are obtained as simple integrals of f . The
individual terms in the sums of (6) are also polynomials in T , and the desired trace formula, in which

T does not appear, is obtained by observing that the constant term of the sum is the sum of the constantterms. To be more precise, the constant term of a polynomial is its value at the origin, so that to specify

Introduction to the work of James Arthur

9

it an origin has to be chosen. Arthur defines his preferred origin T0 in [10], in which as well as in [4]some geometrical principles basic to many of his later arguments are introduced.

I
T

Figure 1

The truncation is particularly simple when a is of dimension one. Then T varies on part of an affine
line and to establish that a function j(T ) of T is a polynomial we need only verify that j(T ) − j(T )is of the form λ(T  − T ), where λ is linear. The main term, thus the kernel itself, does not survive
when the difference is taken; only the truncating terms do. If T  − T lies in the positive chamber, thenaccording to (3) the difference between the two truncated kernels is(9)

KP (δx, δx)σ(H(δx))

P (Q)\G(Q)if σ is the characteristic function of the interval [T, T  ).
The integral over the diagonal of (9) is


(10)



−1

M (Q)\M (A)1 M (Q)

if

fM (m

T

 
fM =
K

  T 

γm)dm
dH

N (A)

f (k −1 mnk)dn dk.

Although we have been betrayed by our assumption that the algebraic group G was semisimple, theformula (10) shows that the difference between the two truncations is a linear term in T  − T , namely

 T
dH,
T

in essence just the length of the interval from T  to T , times a factor that does not depend on T or T  .
Since M is not semisimple and indeed contains a split torus the quotient M (Q)\M (A) is not of finite

Introduction to the work of James Arthur

10volume. To have a group of finite volume, we can replace M (A) by M (A)1 , the kernel in M (A) of themap HM . Since the quotient is then even compact, the spectrum is discrete and the first factor is thetrace of fM .
In addition to the simple roots ∆P of a in the Levi factor of the parabolic P , there is also the set∆P of simple roots of aP in the algebraic group G, and more generally, whenever P  ⊂ P , sets ∆PP  ofthe roots of P  ∩ M , M = MP , in M . There are natural decompositionsaP  = aPP  ⊕ aP ,P
so that we can regard the elements of the basis {α } (of the dual of aP
P  ) that is dual to ∆P  as functions

on aP  . Then the truncations are defined by the characteristic functions τ̂PP of


(11)


H | (H) > 0 ∀ α ∈ ∆PP .

Indeed, for G itself only P = G is used, and then it is dropped from the notation. The other possibilities
are included only because they appear in arguments by induction, of which there are many. They willcertainly not be presented here. All we want is some feeling for the geometry.
If the dimension of a is two, there are four of these functions. For P  = G, τ̂G is identically one,as there are no conditions in (11). If aP  is of dimension one, then τ̂P  is the characteristic function of ahalf-plane and if P  is the minimal parabolic then τ̂P  is the characteristic function of a cone, the cone(I) of Figure 1.
Although the various terms of the truncation are defined by different parabolics and therefore
different kernels, these kernels have similar asymptotic behaviour so that what in effect is happening isthat terms are being subtracted that correspond to the two half-planes that intersect in I and that thena term supported on I is added, so that we are left with a term concentrated on the shaded region, acone extending to infinity. Although itself unbounded, the shaded region defines a bounded region in
the fundamental domain because its intersection with the cone (2) is bounded.
To understand what is obtained from the difference of two truncations, we translate the diagramof Figure 1 from T to T  . The principal term, that corresponding to P  = G, is removed. The
term corresponding to an intermediate P  , with dim aP  = 1, remains, but has been truncated bymultiplication by −1 times the characteristic function of an infinite band of finite width, whose twosides are determined by the projections of T and T  on aP  . For the minimal P  , the difference
corresponds to a truncation by the full strip of Figure 2, itself the union of the three pieces, I, II andIntroduction to the work of James Arthur

11

III. The truncation by I combines with the band for one of the intermediate parabolics P  and yields an
expression that is defined by a kernel for M  , and that therefore can be treated inductively, the width of
the band appearing as an additional factor that is linear in T and T . The truncation by III corresponds
to the second intermediate parabolic. Finally, the truncation by II yields the trace for a kernel on theLevi factor of the minimal parabolic multiplied by the area of II, which is a polynomial in T and T .

I

T´
T

II
III

Figure 2

The argument is less transparent when T  − T is not in the cone (2) but the conclusion is the same,and applies not only to the complete kernel but to the partial kernels defining the geometric and thespectral terms. Until now, we have only needed the parabolic subgroups over Q containing a fixedminimal one, but for many purposes it is necessary to introduce the finite set P(M ) of those for whicha given M is a Levi factor. All those M that contain the originally chosen A need be considered, so that
we are enlarging the set of P to be considered and the set of M . The Q-parabolic groups containing M
can be parametrized by the chambers in aM (or in its dual a∗M ), where aM = aP for any P ∈ P(M ).
A simple, but important, notion appears in (10), that of a (G, M ) family. It is a collection {cP (λ)} ofsmooth functions on ia∗M with the property that cP (λ) = cP  (λ) whenever P and P  are associated toadjacent chambers and λ lies in the hyperplane defining the common wall of these two chambers. The
simplest example of such a family is a collection {eλ(XP ) }, where {XP } is an AM -orthogonal family.
This means that XP − XP  is orthogonal to the wall separating P and P  whenever P and P  are
adjacent. If XP − XP  points from the chamber associated to P to that associated to P  , then the family
is said to be positive. A typical example of such a family is given in Figure 3, where the points of an

AM -orthogonal family appear as well as the convex set that they span.
There is a basic principle that appears in [10]. If {cP (λ)} is a (G, M )-family, then(12)cM (λ) =cP (λ)θP (λ)−1P ∈P(M )can be extended to a smooth function on ia∗M . The function θP is, apart from a constant, the productof linear functions given by the simple coroots. For example, if the family {cP (λ)} is given by anIntroduction to the work of James Arthur

12

AM -orthogonal family, then cM (λ) is the Fourier transform of the characteristic function of their hull,so that the value cM of cM (λ) at the origin is its volume.

X P´
XP

Figure 3

The sum of two AM -orthogonal families is again an AM -orthogonal family. More generally, theproduct of two (G, M )-families {cP } and {dP } is a (G, M )-family {eP }. It is very important that eM ,or even eM (λ), can be expressed in terms of functions associated to {cP } and {dP }. This is the sourceof a splitting principle that, among other things, reduces the global contributions to the trace formula
to a sum of products of local contributions. If, for example, the two families are given by positive

AM -orthogonal families {XP } and {YP }, so that {eP } is given by {XP + YP }, then eM is a volume. In
Figure 4 the area of the darkly shaded region is cM , the areas of the six lightly shaded regions add up todM . The regions themselves are parametrized by the parabolic subgroups with M as Levi factor, thus
by the elements of P(M ), and the area is given by a number dP , which is the value at λ = 0 of a smoothfunction dP (λ) given by a formula very similar to (12). Such a function is defined for each parabolicQ
Q. Each of the remaining regions has area equal to cQ
M dQ where Q is an intermediate parabolic, cM isdefined relative to a Levi factor of Q (rather than relative to G itself). The general formula reads(13)eM (λ) = Q
cM (λ)dQ (λ),
Q

where the sum is over the set of parabolic subgroups containing M .

Introduction to the work of James Arthur

13

XP+YP
XP

Figure 4

3. In pursuit of a trace formula. As was emphasized, the formula (6) is not invariant and contains,in addition, an undesirable dependence on the variable T . The dependence on T has been discovered
to be weak and has been eliminated by setting T = T0 , a simple, but persuasive, device as all terms on
both sides are polynomials in T whose higher order terms are of no intrinsic interest because they areassociated to Levi factors of proper parabolic subgroups of G.
Although there are circumstances (cf. [K1]) in which the trace formula simplifies on its own,
in general it is best to work with an invariant form. The invariant form has an appealing structural
simplicity, but the apparent disadvantage that the geometric sides contain hidden spectral information.
It is

(14)

 WM
M



 WM 0aM (γ)IM (γ, f ) =W0G
Mγ∈M (Q)

0

W0G

Π(M )

aM (π)IM (π, f )dπ.

The sum, taken from [26], is over the Levi factors containing the fixed A. The factors W0M are orders of
Weyl groups. So it is the other terms that need to be understood. The expression on the right anticipates
the resolution of a problem of convergence, partially dealt with by W. Müller in [Mü]. In the formula,as it stands to date, some attention has to paid to the order in which the sum and the integral on theright are taken ([27]). The left side too is simpler in [26] than in [27]. The factors aM (γ) are in reality
factors aM (S, γ) that, for reasons that will appear later, depend in addition on the choice S of a large
finite set of places.

Introduction to the work of James Arthur

14

Although the basic technique for creating invariance was introduced in [10], the final form does
not appear until later, in [26], [27]. In particular, an important local principle, a Paley-Wiener theorem,
was needed but not proved until later. It is a theorem for real groups of considerable interest in its own
right, but also essential for rendering the spectral side of the trace formula at all explicit as in [13] and[15], and this was apparently the original purpose of the theorem of [15]. The Paley-Wiener theoremon the real line is well known. It characterizes the Fourier transforms of functions (or of smoothfunctions) with compact support as entire functions with supplementary properties. Arthur’s theorem
characterizes the Fourier transforms of smooth, compactly supported functions f on a reductive or
semisimple real group whose translates, on the left and right, under a maximal compact subgroup
generate a finite-dimensional space in terms of the behaviour of the operators
π(f ) =

f (x)π(x)dx,

as π runs over the representations I(σ, λ) induced from representations σ ⊗ eλ(H) of the Levi factor of aminimal parabolic subgroup of the real group. Since the induced representations have a structure that
is, at best, difficult, the characterization is not at first glance very useful, but a simple easily applicableconsequence is deduced from it in [15] in the form of a multiplier theorem, that provides a large familyof functions Γ(σ, λ) such that when I(f, σ, λ) is defined by an element of the Paley-Wiener space, thenso is Γ(σ, λ)I(f, σ, λ).
Arthur’s characterization is not, however, adequate to the arguments of [10]. For that anothercharacterization, not of the family of operators π(f ) but of their traces and for tempered representations

π alone, that is due to Clozel and Delorme [CD] is necessary.
The truncation used to obtain a trace formula was a truncation on the diagonal tailored to thegeometric side. Although it was applied to the partial kernels Kχ (x, y), it is not adapted to them,as they are defined by a decomposition of L2 (Γ\G). An important step at the very beginning of thedevelopment of the spectral side is to show that there is a geometrically defined projection operatorΛ = ΛT on L2 (Γ\G) such that the integrated truncation is also the integration (over the diagonal) of
the kernel of ΛK and the individual terms, therefore the integration of the kernel of ΛKχ . Of course

ΛK can be replaced by KΛ or, as Λ is a projection, by ΛKΛ. The operator Λ does not commute withthe action of the group G.
In order to understand the implications of this replacement, we recall – for the first time – thedefinition and the structure of the spaces L2χ (Γ\G). The class χ is defined by a pair (M, ρ), but
to avoid an excess of subscripts and superscripts we take it as given and free M for other uses.

Introduction to the work of James Arthur

15

The spectral decomposition of L2 (Γ\G) as a whole and of L2χ (Γ\G) in particular is given, as is any
spectral decomposition, by functions depending on the spectral parameters. Since it is the spectral
decomposition of a group representation and not that of a single operator, these parameters include
not only π but a second parameter φ, an element of the vector space on which the group acts. In thecase under consideration, the functions E(x, π, φ) are called Eisenstein series. The first step in theirdescription is to use the collection χ to classify, for each possible Levi factor M , the discrete spectrumof M (A) in L2 (M (Q)\M (A)1). We have to pass to M (A)1 and not remain with M (A) because theinevitable presence of the centre AM of M when M = G prevents M from having a true discrete
spectrum.
An irreducible unitary representation π of M (A) determines by restriction one of M (A)1 , two
representations π and π determining the same restriction if and only if

π  (m) = eiλ(HM (m)) π(m),for some element λ of the real dual a∗M of aM . Conversely, if π is a representation of M (A)1 and λ ∈ a∗Mthen π can be extended to a representation πλ of M (A) by writing m as the product of eH , H ∈ aM ,and m in M (A)1 and then setting(15)

πλ (m) = eiλ(H) π(m ).

A Lebesgue measure on the parameter λ together with the uniform atomic measure on the unitary
representations of M (A)1 provides a measure dπ on the set of unitary representations of M (A). To
each χ is associated a subspace

L2χ (M (Q)\M (A)1) =HM,χ (π)π

of L2 (M (Q)\M (A)1), the sum running over the irreducible unitary representations of M (A)1 . The
space HM,χ (π) is a space that decomposes as a finite direct sum of irreducible representations allequivalent to π . Moreover, it will be 0 except for a countable number, even in general a finite number,of π . (In order not to encumber the language, we shall frequently refer to the representation on this spaceas π , thus suppose, for simplicity, that the multiplicity is one, as is often the case.) Spaces HM,χ (πλ )are then defined by the construction (15). The Eisenstein series E(x, φ) are attached to elements φ inHM,χ (πλ ) and to a parabolic subgroup P with M as Levi factor. Then the family of functions

x→

Π(M )

EP (x, φ(π))dπIntroduction to the work of James Arthur

16(where of course π → φ(π) is subject to constraints on its integrability, on its smoothness or on its
support) generate a subspace L2P,χ (Γ\G) of L2χ (Γ\G). A simple equivalence relation, that of being
associate, is introduced on parabolics and

L2P,χ (Γ\G) = L2P,χ (Γ\G)
depends only on the class P of P . The result is that

L2χ (Γ\G) =

(16)



L2P,χ (Γ\G).

P

The spectral dimension of the component L2P,χ (Γ\G) is dim aM if M is the Levi factor of P ∈ P.
As a consequence of the formula (16) the kernel Kχ can be expressed as an integral over theparameters P and Π(M ) of Eisenstein series
P P ∈P

1
n(A)


Π(M )

E(x, φj (π))Ē(x, φk (π))dπ,and the truncation ΛKχ Λ as
P P ∈P

1
n(A)


Π(M )

ΛT E(x, φj (π))ΛT Ē(y, φk (π))dπ.

The family {φj (π)} is an orthonormal basis of the space of π , usually chosen so as to vary in a coherent
manner. Thus the integral of the truncated kernel ΛKχ Λ isJχT (f ) =

(17)


P P ∈P

1
n(A)


Π(M )

tr(ΩTχ (P, π)ρχ (P, π, f ))dπ,

where the operator ΩTχ (P, π) is so defined that
(18)
G(Q)\G(A)ΛT E(x, φj (π))ΛT Ē(x, φk (π))dx = tr(ΩTχ (P, π)ρχ (P, π, f )).

I recall that the integral over Π(M ) in (17) is a Lebesgue integral over the dual of AM = AP followed
by a discrete sum. The notation is not exactly that of Arthur, for his emphasis is different than ours,
but the comparison with papers [12], [13], [14] should not be difficult.
Such abstract considerations can hardly be very meaningful. If it is noticed that the class χ has arank, the dimension of the centre of the Levi factor to which it is attached, it is seen that the spectral
dimensions of the components of (16), thus the dimension of A = AP , vary from 0 to this rank. It may
happen, as in the unramified case appearing in formula (8), that only the largest of these dimensions

Introduction to the work of James Arthur

17

appear. That is the favorable, or at least the simplest, situation, for one of the difficulties that plague
Arthur’s development of the trace formula, a residual effect of the methods used to establish the theoryof Eisenstein series, is that the spectral side is only a sum over χ so that there is no clean separationof the individual terms into spectra of different dimension and no guarantee that such a separation ispossible without destroying convergence.
In order to reduce the level of abstraction and to introduce some concrete formulas, I shall examine
the case that the rank of χ is one, adequately represented, both structurally and analytically, by thegroup SL(2). With some understanding of this case and the standard analytic techniques on theline used to analyse it in hand, we shall then try to appreciate the new type of analysis that Arthur
substituted for them in higher rank.
Before beginning, we recall from [9] a formula not for (18) but for a perturbation of it. It is valid
only when the Eisenstein series are associated to cusp forms, thus when the the dimension of AP isequal to the rank of χ. We fix a given π and consider all the other defined by

πλ (m) = eλ(H(m)) π(m).

(19)

This notation differs slightly from that of (15). We replace (18) byΛT E(x, φ, λ)ΛT Ē(x, ψ, −µ̄)dx,(20)
G(Q)\G(A)in which E(x, φ, λ) is just E(x, φj (πλ )), but in which, for the next formula, it is best not to assume thatµ = −λ̄. By the definition, the space of πλ is the same as that of the given π used to define it. So the
Eisenstein series are parametrized by elements of a common space and by λ. If we let λ approach −µ̄in (20) we recover (18).
For an arbitrary π , the theory of Eisenstein series associates intertwining operators M (t, π) toappropriate elements of various Weyl groups. Since we are concerned with the behavior of M (t, πλ )as a function of λ, we write M (t, λ) rather than M (t, πλ ). The formula for (19) isΛT E(x, φ, λ)ΛT Ē(x, ψ, −µ̄)dx =(21)
G(Q)\G(A)  (M (s, λ)φ, M (t, −µ̄)φ )Ps,tθP (sλ + tµ̄).

The function θP has already appeared in (12). The sums over s and t are finite sums, and s and t may,without much loss of precision, be thought of as elements of a Weyl group.
Although it does not simplify, the formula (21) becomes more transparent and the limit λ → −µ̄easier to take when the rank of χ is one and, therefore, the cuspidal Eisenstein series functions of a

Introduction to the work of James Arthur

18

single complex variable. The dimensions of the spectrum appearing in (12) are then one and zero,
the zero-dimensional spectrum being of course the discrete spectrum. Moreover, the variable λ canbe identified – not canonically – with a complex number iλ = σ + iτ . In (19) and (21), we are at firstpreoccupied with purely imaginary λ, for they yield the one-dimensional spectrum. Set

Θ(τ ) = e−2iτ T M (iτ ).

(22)Then, as in [L], the formula (21) becomes in the limit λ = µ, σ = 012− Θ−1 (τ )Θ (τ )φ, ψ) + (Θ(τ )φ, ψ) − (φ, Θ(τ )ψ) .
i
iτ

(23)

It can happen – this is basically the unramified case – that there are very few s and t in the sum (21) sothat the second term does not appear. Then the following considerations simplify.
We have noted how important it is for the further development of the trace formula that the left
side of (17), and thus also the right side, is a polynomial in T , indeed, when the rank of χ is one, apolynomial of degree one. This ceases to be so if we separate the terms of various rank. The terms ofrank one are obtained by taking φ and ψ in (23) to be functions of τ and then integrating over (−∞, ∞).
The first term, a logarithmic derivative, is what appears in the unramified case and clearly yields a
linear function of T . The second term, in its dependence on T is essentially an integral with kernelsin(τ T )/τ , so that as T grows larger and larger it will just pick up the value of whatever it multiplies at
τ = 0. Thus it will approach a constant, but there will be a remainder which is not a polynomial in T ,but is very small, at least for T large. Since the complete sum (17) is a polynomial, this small remainder
will be cancelled by a small remainder from the discrete spectrum.
The discrete spectrum is also obtained from the Eisenstein series, as functions of x ∈ G obtained
as the residue of E(x, φ, λ) at certain real points λ > 0. We can readily obtain the inner product of two
truncated residues by taking residues in the formula (21). As in [L], it will have, apart from a constantfactor, the form

(24)(m(λ)φ, ψ) −1
e−(λ+µ)T (m(λ)φ, m(µ)ψ)).
λ+µThe first term is a constant, thus a linear polynomial in T ; the second is small for large T . So the resultthat (16) is a polynomial in T is compatible with these formulas. Moreover, we now have a little more
information for χ of rank one. Up to remainders that become small as T grows large, individual termsin (16) are polynomials in T for which there are explicit expressions in terms of logarithmic derivativesof the operators M (t, π).

Introduction to the work of James Arthur

19

These results are established in general in [13], [14]. In rank one, the constant term is the sumof a one-dimensional logarithmic integral, the δ -function generated by the sin(τ T )/τ kernel, and thecontribution from the first term in (24), which is nothing but the inner product of two nontruncatedsquare-integrable automorphic forms in the discrete spectrum. In general, each term of (17), except for
those of rank zero and of the highest rank, will combine all of these features. So I refer to Theorem 8.2
of [14] for their precise description. On the other hand, I would like to comment briefly on the proof,
since it illustrates Arthur’s ability to overcome difficult analytic obstacles with unexpected extensionsof traditional analytic techniques. I fear that, imbedded as they are in developments that also carrya heavy algebraic and even arithmetic burden, traditionally oriented analysts have not given thesecontributions the attention they deserve.
His first step ([12]) was a form of (21) valid for Eisenstein series associated to all elements of thediscrete spectrum and not just the cusp forms. It is suggested by (24), in which only the first term
counts. It turns out that in general the inner product of two truncated Eisenstein series is the sum of aterm for which there is an analogue of (21) and a term that grows exponentially small as T grows large.
A simple case appears in the formula (24).
This formula yields an asymptotic expression for the functions appearing in (17)ΨTπ (λ, f ) =1
tr(ΩTχ,π (P, λ)ρχ,π (P, λ, f )),n(A)where, once again, as in (19) the notation is adapted to a fixed π and a varying πλ . It is tempting toinsert this asymptotic formula directly into (17) and to ignore the small terms. The difficulty is that
there is no uniformity in λ that permits an integration or the discard of the small terms.
So a cut-off function in λ with compact support on which it is largely one or close to one has to be
introduced in the integrals of (17) and then dilated so that the result approaches the original integral.
As a result of this modification, the expression is no longer a polynomial in T , because the cut-off is nota result of modifying f . (Cutting off the Fourier transform of a function with compact support does
not yield the Fourier transform of a function with compact support!) What Arthur does in [13] is to
exploit the multiplier theorem resulting from his Paley-Wiener theorem to show that the cut-off is still
approximately a polynomial as T grows and that, in addition, as the cut-off function approaches in asuitable sense the constant 1, this polynomial approaches (17). The possible failure of the Ramanujan
conjecture creates an additional difficulty in the analysis, but once it is accomplished he is free to use
his approximate formula.

Introduction to the work of James Arthur

20

He has then to extend the calculation leading to (24) from dimension one to arbitrary dimension.
The functions that result from the general form of (21) are, like (22), products of exponentials andintertwining operators and appear in the form of (G, M )-families. The sin(τ T )/τ kernel is essentiallythe Fourier transform of the characteristic function of the interval (−T, T ) and its properties can readily
be deduced by passing to Fourier transforms. This interval is replaced in general by the convex sets
described in §2, so that the sin(τ T )/τ kernel is replaced by a function of the form (12). The formula(13) allows him to treat the exponential factors separately from the intertwining operators.
This is by no means the only application of (13). An automorphic representation π can be expressedas the tensor product ⊗πv over the places finite and infinite of the field F in question (for us F has
been Q) of representations of the local groups G(Fv ). There is a corresponding factorization of theintertwining operators M (s, π) or, if π is replaced by πλ , M (s, λ). The local operators are themselves
products of a scalar that depends on λ and a second simple factor that is almost everywhere 1, so that thetensor product of the second factors is of more algebraic than analytic interest. The product of the scalar
factors is of analytic interest and turns out to be an L-function, classical and equal to the ζ -function or
to a Dirichlet L-function in the simplest cases but a more general Euler product otherwise. Once again,
it is formula (13) that enables Arthur to separate these two factors. Although this separation is of major
importance when applying the trace formula and is used on the way to (14), its immediate relevanceis that it yields an estimate that allows a passage to the limit in the mollifier after the approximate
formula for ΨTπ (λ, f ) have been inserted. The result is a decisive step on the way to the right side of
the compact formula (14) in which that of Theorem 8.2 of [14] is implicit. This theorem is the precise
general form of formula (8). We have yet to consider the precise general form of (5).

4. Weighted orbital integrals: reduction to local integrals. If we consider the truncations only
for T = T0 , it is only the weight v(x, T0 ) that matters in (5). It can be expressed as vM = vM (x),where vM (x) is defined by a (G, M )-family {vP (λ, x)} associated to an AM -orthogonal family thatis introduced by observing that the functions HP (x) described earlier for the Q-parabolic subgroups
containing a given minimal one can be defined more generally and that for each x the family {HP (x)}is AM -orthogonal.
Although we have not yet undertaken the comparison of trace formulas, the purpose of suchcomparisons is usually an equality between part or all of the spectral sides of the trace formula for
appropriate functions f and f  on two different groups, an equality that results from an equality on
the geometric side, in turn an equality, at least in part, between the local orbital integrals of f andIntroduction to the work of James Arthurf  , usually taken as products f (x) =21

fv (xv ), f  (x ) =fv (xv ). If this strategy is to be used, thegeometric side of the trace formula has to be expressed, perhaps with the help of the splitting principle,
in local terms. The expression (5) is not local, but it has a local form. The one preferred by Arthur is
not an integral over 2 Gγ (Fv )\G(Fv ), but one over Gγ (FS )\G(FS ), where S is a finite set of places,
usually large and usually containing all infinite places. In addition he multiplies it by a simply defined
factor|D(γ)|1/2 =|detg/gσ (1 − Ad(σ))|v1/2 ,v∈Swhere σ is the semisimple part of γ . Thus when γ itself is semisimple, as in (5), σ = γ . The result isthat (5) is replaced by

JM (γ, f ) = |D(γ)|1/2(25)
Gγ (FS )\G(FS )f (x−1 γx)vM (x)dx,

an expression that is defined for all γ in M (FS ) whose centralizer Gγ in G is equal to its centralizer

Mγ in M . The element γ need no longer be semisimple and Gγ (FS ) is really a productGγv (Fv ).

Because of the splitting principles deduced from (13), it will also be appropriate to consider (25) relative
to a Levi factor intermediate between M and G, but that is a formal matter. Although the weighted
orbital integrals (25) appear first in connection with the unramified classes on the geometric side, thenext step in the development of the trace formula is to define them for all classes and to express eachsummand on the geometric side in terms of them.
Arthur’s treatment of the geometric terms appears in [20], [21], [24]. One part, carried out in [21],is simply to analyse the combinatorics of the truncation to reduce the geometric term associated to a
general orbit o to the term associated to the unipotent orbit o = unip. Beyond that, there are two things
to do: to define the distributions JM (γ, f ) in general, thus for arbitrary γ ∈ M (FS ) and to express

Junip in terms of them. These are the tasks of [24] and [20] respectively. The result is a formula ([21],Theorem 8.1) that looks just like a sum of some of the terms on the left of (14), except that the terms
are not yet invariant. Rather than try to understand immediately how this procedure is carried out ingeneral, we examine the simplest case, that of PGL(2), observing that it suffices to define JM (γ, f )when S consists of a single place.
The matrix


γ=

1 1
0 1



2 At this point, it becomes more convenient to work explicitly with a general number ﬁeld and not to reduce allquestions to Q.

Introduction to the work of James Arthur

22

represents a typical unipotent element and D(γ) = 1. Any element of G(Fv ) can be written as x = nakwithn = n(z) =

1 z
0 1


,

a = a(α) =

α 0
0 1

and with k in the standard maximal compact subgroup. Thendx = dz

dα
dk.
|α|2v

Thus, apart from some simple constants that are overlooked in the following sequence of equalities,
JG (γ, f ) =

(26)

=

Gγ (Fv )\G(Fv )



dk


=


dk

f (x−1 γx)dx

dα{f (k −1n(1/α)k)}|α|2vdz{f (k −1n(z)k)}.

The final integral is clearly convergent if, for example, the support of f is compact. If, on the other
hand, γ = 1, then

JG (γ, f ) = f (1).
There is only one more unipotent weighted orbital integral in addition to this.
Suppose P is the group of upper triangular matrices and P  the group of lower triangular matrices.
Their common Levi factor M is the group of diagonal matrices. The remaining weighted unipotent
orbital integral is JM (1, f ). According to the procedure introduced in [24], it is introduced as a limit.
The matrix n = n(z) is in P , so that HP (n) = I , and ifn(z) =

α
0
0 α−1



1 0
k,
y 1

then, as we are in the projective group, HP  (n) = a(α2 ), α being easily calculated. As a result – a factor
depending on the normalization of measure aside – the weight vM (n) = vM (nk) is ln(1 + z 2 )/2 if v isreal and if v is nonarchimedean it is 0 for z integral and ln |z| if |z| > 1.
Consider JM (a(α), f ) for α close to 1. It is|α − 1||α|1/2or|α − 1||α|1/2




dk




dk

dz{f (k −1n(−z)a(α)n(z)k)vM (n(z)k)}dz{f (k −1 n((α − 1)z)a(α)k)vM (n(z))}.

Introduction to the work of James Arthur

23

The integral over k is easily dealt with as vM (n(z)k) = vM (n(z)). We replace the variable z by z/(α−1)
and let α approach 1. The denominator |α|1/2 of |D(γ)|1/2 becomes 1 and the numerator is cancelledby the change in measure. If F is the real field, the term vM (n)(z/(α − 1)) isln((α − 1)2 + z 2 )/2 − ln |α − 1|.
The second term leads to a contribution that blows up as α approaches 1, although in a simple way,
for the contribution is the product of a term in α that is independent of f with (26). The first term
approaches, except at z = 0, the function ln |z|. We set(27)

JM (1, f ) =dz{f (k −1n(z)k) ln |z|}.

dk

A similar calculation leads to the same result for nonarchimedean fields.
Thus, for γ = 1, we have defined JM (γ, f ) as a limit

lim

a→1



L
rM
(γ, a)JL (aγ, f ),L⊃M

where a lies in AM (F ), provided we define (constants aside)GrM(γ, a) = ln |α − 1|.

MrM(γ, a) = 1,

The sum runs over those Levi factors of parabolic subgroups that contain M , in the present case just G
and M itself. The purpose of [24] is to do this for all groups G, all M and all γ ∈ M (F ).
If we pass to FS , then the factor ln |z| in (27) is replaced by

ln |zv |v .

v∈SIn the sequence of modifications (26), an integral over the multiplicative group {a(α) | α ∈ F × } isreplaced by an integral over the additive group {n(z) | z ∈ F }, z = 1/α. If we had naively attempted
to compute the integral of the kernel K1 without truncating, then in addition to the contribution from


(28)

Γ\G

f (1)dx,

we would have, with γ = n(1)(29)meas(Gγ (Q)\Gγ (A))

Gγ (A)\G(A)f (x−1 γx).

Introduction to the work of James Arthur

24

Since Gγ = N , we can replace this integral by an integral over A(A) × K , where A is the groupof diagonal matrices in G and K is the maximal compact subgroup of the global group G(A). We
might attempt to apply the sequence of transformations (26), but we would run into a difficulty,
and this difficulty is the principal reason for the truncation. If v is a nonarchimedean prime, let

A0 (Fv ) = {a ∈ A(Fv ) | |a|v = 1}. Since A(A) is the limit of the sets


US =



A(Fv )A0 (Fv ),

v ∈S
/

v∈S

where S is finite and contains all infinite places, we can first try to calculate the integral over A(A) as
an integral over US . If S is so large that outside of S the function fv is the characteristic function of thegroup Kv of integral matrices with integral inverse, then the integral is a product of an integral over

A(FS ) with the integrals
A0 (Fv )

1=

dα
.
2
|α|v =1 |α|

The global integral can and must be defined by normalising the multiplicative measure so that each ofthese integrals is 1. Suppose F = Q and v is the prime p. If we tried to pass to the additive measure,taking z = 1/α, then we would have1dz = 1 − ,p|zv |=1

the equality resulting from the necessary normalization of the additive measure. So the transition
from the penultimate line of (26) to the final line would involve not a harmless finite constant but ameaningless infinite product



1
1−
.
p

p∈S
/

If, however, we had truncated, then the integral would not be over A(A), but only over thoseelements a(α) in this group for which |α| < c, c = cT being a constant that depends on the truncation
parameter. Fix once again S and suppose once again that fv is the characteristic function of Kv outsideof S . Then for each {αv | v ∈ S}, we would only have to integrate over the collection of

αv | 1 ≤ |αv |v

and

v ∈S
/


v ∈S
/

|αv |v ≤ c/v∈SWhen F = Q so that each of these places is a prime p, this integral is

(30)

 1
n≤C

n

,


|αv |v .

Introduction to the work of James Arthur25

the sum being taken only over those n that are not divisible by any prime in S . The number C isc/v∈S |αv |v . It is well known that the sum (30) is asymptotic toa + b ln C = a + b ln

(31)



|αv |v = a − bln |zv |v ,v∈Sif zv = 1/αv . We conclude that (29) can be expressed as a sum of multiples of JG (γ, f ) and JM (1, f ),the coefficients depending on the choice of S , which must be taken sufficiently large to accomodate thegiven function f .
The constants a and b clearly depend on S . (The constant a also depends linearly on T , but b
is independent of it.) When describing the invariant form of the trace formula, I cautioned the reader
that the coefficients aM (γ) depended on the choice of a set S . Since the invariant geometric side withthe distributions IM (γ, f ) is obtained from the geometric side with the distributions JM (γ, f ), we
see clearly, in the simple case just treated, that the source of the dependence on S is the asymptotic
behaviour of (30).
The argument in general is similar, but the estimates more difficult, and the measures not directly
identifiable. The conclusion is a formula,(32)

Junip (f ) = |W M | 0M|W0G |aM (S, u)JM (u, f ),u

in which the outer sum is over the Levi factors of the parabolic subgroups. The inner sum is over unipotent u with rational representatives. A decisive observation of [10], a paper to which we shall return, isthat the distributions JoT (f ) and JχT (f ), although not invariant, have similar formal behaviours under
the similarity transformations, f → f y , f y (x) = f (yxy −1 ). Many other distributions that appear in theanalysis of the trace formula behave in the same fashion. In particular, the weighted orbital integrals

JM (γ, f ) do. As a result, it is possible to define inductively coefficients aM (S, u) such that
Junip (f ) − |W M |0I(u, f )aM (S, u)JM
G
|W0 |
M =G

is invariant. It is also supported on the unipotent classes, so that once it is shown that it is a measure, it
is clear that it is the sum of multiples of the distributions JG (u, f ), u unipotent. If these u can be shownto have rational representatives, the formula (32) follows. What is required is that sums similar to (30)but much more complicated be estimated with conclusions much coarser than (31).

5. The invariant trace formula: introduction. The paper [10] was Arthur’s first attempt at aninvariant trace formula. It was not complete as it invoked some assumptions that were only verified

Introduction to the work of James Arthur

26

later, but in addition to the geometric ideas already described in §2 it contains several ideas that reappear
as a rogue’s yarn throughout the later work.
I have already alluded to the formal structure of the noninvariance of the distributions JoT andJχT and thus of Jo and Jχ . We have already agreed that the only parabolic subgroups to be considered
are those that are defined over the ground field F , usually Q, and that contain a fixed Levi factor M0of a fixed minimal parabolic P0 over F . The only Levi factors to be considered are Levi factors of thiscollection of parabolic subgroups that contain the fixed Levi factor M0 . The symbol LL (M ) denotes
the set of Levi factors between two Levi factors M ⊂ L; the symbol FL (M ) the parabolic subgroups
over F between M and L; and PL (M ) those with M as a Levi factor. If L = G, the superscript is often
omitted. If Q lies in FL (M ), then as in [10] we associate to the function f a function on MQ (A) by

 
(33)

fQ,y (m) = δQ (m)K

NQ (A)f (k −1mnk)uQ (ky)dn dk,

where the function uQ is defined by the geometric constructions of §2. It is a volume! If f y (x) =f (yxy −1 ), then(34 )JoT (f y ) =



M ,T

Jo Q (fQ,y ),Q

the sum running over all allowed F -parabolic subgroups and the double superscript indicating clearly
that the distributions on the right are on the group MQ . Moreover(34 )JχT (f y ) =JχMQ ,T (fQ,y ).

Q

Thus the formal structure is the same for both the geometric and the spectral distributions, and isindependent of T . The definition (33) admits a local form, and the weighted orbital integrals JM (γ, f ),which are local objects, satisfy a relation just like (34 ) or (34 ), in which, however, the sum is over thoseQ that contain M and are defined over the local field. There are Q that are defined over a local fieldFv but not over the global field F . Another manifestation of the difference between the local field andthe global field is that the space aM may grow when one passes to the local field so that the geometry
used to define (G, M )-pairs changes.
The method used to convert a distribution satisfying the relations (34) to an invariant distribution
is at first far from promising, as it amounts to little more than transferring the offending part of one
side of the formula, usually of the spectral side, to the opposite side. It is a construction intermediate
between local and global, thus a construction over some product FS =S Fv .

So it is understood

Introduction to the work of James Arthurthat f =27

fv , where fv is the unit element of the Hecke algebra outside of S . Thus all that matters is

S fv , which we also denote f and which we regard as an element of an appropriate space U (G). The

argument proceeding by induction, we introduce similar spaces U (M ) on all pertinent Levi factors.
There is a second space V (M ) and a map

φM
M : U (M ) → V (M )through which all invariant distributions uniquely factor. More precisely, the map is surjective andpull-back on the dual space to V (M ) has as image the set of invariant distributions on U(M). Moregenerally, the maps φL
M : U (L) → V (M ) are to be defined for every pair M ⊂ L of pertinent Levifactors and the relations3yφLM (f ) =(35)

M

φMQ (fQ,y )Q∈F L (M )are to be satisfied.
These objects available, we can pass from a family of distributions J L , one for each Levi factor L,
that satisfy 4

J L (f y ) =
Q∈F L (M )c(MQ ) MQ(fQ,y )Jc(L)to a family of invariant distributions IL by demanding that

J L (f ) =(36)


M ∈LL (M0 )c(M ) ˆM L
I (φM (f )).
c(L)The distribution I M is the image of IˆM . It is defined inductively by this formula.
Arthur passes quickly – and rather glibly – in [10], returning to the matter in [26], [27], [28], fromFS to Fv , on the grounds that if these assumptions are satisfied for the maps φL
M that he constructs
when S consists of a single place v alone then a product over v in S yields them in general. At all
events, the simpler case is the decisive case. The definition of φL
M is based on the local analogue of theintegrals appearing in (17).

6. Weighted characters and the invariant trace formula. In this section, we at first fix the place v
and let F denote the local field. The question as to which space to take for U (G) is not fully resolved
3 For technical reasons, this relation must sometimes take a diﬀerent, integrated form. The path to the invariant
trace formula is, in contrast to others more heavily trodden, not yet broad.
4 I have included in the formula constants c(L) attached to Levi factors because Arthur does, but they are often
identically 1.

Introduction to the work of James Arthur

28

until the papers [27], [28]. Although there is an important technical complication – to be explained
when we are in a position to appreciate it – that imposes a modification to the definition, it is essentiallythe Hecke algebra of smooth, compactly supported functions that are left and right K -finite, K being
now a local maximal compact subgroup. The space V (G) is defined by the Plancherel formula. As is
known from the work of Harish-Chandra, the representations needed for the local harmonic analysisof L2 -functions or of Schwartz functions are the irreducible tempered representations. For any G, thespace 5 V (G) is a space of functions on the set Πtemp (G) of tempered representations. The local andglobal harmonic analysis are structurally similar. The spectrum of L2 (G(F )) is described by a sum
over Levi factors (with the same conventions as above, but with respect to a minimal parabolic over F
and a distinguished Levi factor of it), each Levi factor contributing spectra of dimension equal to that of

AM , the maximal split torus in its centre. The components of the contributions of M , apart from some
identifications arising from the action of the Weyl group, are indexed by the representations of M (F )square-integrable modulo its centre, referred to here, by abuse of terminology, as representations of thediscrete series. The definition (19) may be used locally, although I now prefer to add a factor i as in(15),

πλ (m) = eiλ(HM (m)) π(m).
If σ is a tempered representation or more generally any irreducible representation of M (F ) and P aparabolic subgroup with M as Levi factor, there is associated to σ and P an induced representation

IP (σ). The space V (G) is essentially the space of functions on Πtemp (G) such that for any M and any
tempered representation σ of M (F ), the function φ(IP (σλ )) is of Paley-Wiener type. This means inparticular that it extends to an entire function of λ.
G
GSince φL
M is just φM for G = L, it suffices to define φM . It is defined by specifying, for each f , thefunction φ(π), φ = φG
M (f ). As this will be defined by an analogue of the weighted orbital integral, we
denote it JM (π, f ) and refer to it as a weighted character. Then(37)

φM (f )(π) = JM (π, f ).

Because it is easy to forget, when attempting to understand the definitions, that for M = G, whichis the critical case for pulling back distributions, there is nothing to them, I observe in particular that
5 Arthur’s notation develops with time, so that the notation of [10] is not always that of the later papers.
The reader who consults [26], [27], [28] will have to accommodate himself to the newer notation. In this report, I
sometimes adhere to a ﬁxed convention and sometimes, to make consultation of the original papers easier, use two
symbols for what is essentially the same object.

Introduction to the work of James Arthur29

φG (f )(π) = JG (π, f ) = tr(π(f )). The relation (35) becomes the local analogue of the relations (34 )and (34 ) for weighted orbital integrals,

JM (π, f y ) =

(38)



M

JM Q (π, fQ,y ).

Q∈F L (M )With (38) in mind, Arthur defines 6 JM (π, f ) by a local analogue of the integrand appearing in theformula (8) for JχT (f ).
The formula (8) is deduced from (21) in which the global intertwining operators M (s, λ) of thetheory of Eisenstein series appear. These operators have local analogues, denoted JP  |P (πλ ) in [28]that arise because a representation IP (π) induced from the extension of π to the parabolic P with Levifactor M will usually be equivalent to other induced representations IP  (π  ) induced from a sometimes
different π  with a usually different P  but with the same M . The global operators are tensor products of
the local operators, defined directly in a large open cone by an integral and then, in general, by analytic
continuation. It is convenient to represent both as the product of a scalar factor and a normalizedintertwining operator.
The scalar factor is expressed as the quotient of products of automorphic L-functions, local or
global, thus in particular as an Euler product that admits an analytic continuation. Although there is noambiguity about the local scalar factor at those places where no ramification is present, nor in principle
at the other places, at the moment the information available about representations of the groups M (F )is inadequate and some not entirely satisfactory expedients have to be invoked ([28]).
At unramified places, the local normalized operator RP  |P (πλ ) fixes the unramified vector. In
general it is a rational function. The weighted characters are defined by the collection of operators, one
for each P in P(M ),(39)

RP (ν, πλ , P0 ) = RP |P0 (πλ )−1 RP |P0 (πλ+ν ),which turns out to be a (G, M )-family to which the definitions of §2 can be applied7 The groupP0 ∈ P(M ) does not affect the final result. The operator RM (πλ , P0 ) is defined by (12), with cP (λ)
given by (39), the variable λ of (12) being the variable ν of (39) and the λ of (39) being a variable that
6 He later, in the paper [47], slightly modiﬁes the original deﬁnition, which turns out not to be entirely suitable
to his ultimate aims. The original deﬁnition is nevertheless not without its merits. The deﬁnition aﬀects the valuesof the factors aM (π) in (14).
7 The conceptual elegance of the deﬁnitions of §2 is not always matched by a practical simplicity, so thatturning them into explicit formulas does not appear to be always feasible. For the families under consideration, it
is instructive to turn to §7 of [14].

Introduction to the work of James Arthur

30

affects the family cP and thus the value cM . It is well to recall that when the dimension of aM is one,

RM (πλ , P0 ) is going to be a logarithmic derivative. The weighted characters are finally defined as
JM (πλ ) = tr(RM (πλ , P0 )IP0 (πλ , f ))and do not depend on the choice of P0 . The representation π = πλ is unitary but need not be tempered.
Although we did not describe them explicitly either in formula (8) or when mentioning Theorem
8.2 of [14], the global operators that appear are obtained by a global normalization that is a product of
the local normalizations. The global factorization together with the global geometry leads to a global
analogue of the weighted characters. Despite the difference between the local and global aM and theirgeometries, the properties of the global weighted characters are deduced from those of the local. Oncethey are available, Arthur can apply (36), with L = G, to the distributions Jo and Jχ to obtain Io and Iχ .
Then, as a result of easily justified formal manipulations, he obtains ([10]) an invariant trace formula



(40)

Io (f ) =Iχ (f ).

It has yet to be explained why this is the formula (14). Moreover, it has to be decided exactly what
the spaces U (L) and V (L) are to be and whether the maps φL
L are indeed open and surjective, thus
as Arthur later phrases it, in for example [38], whether all invariant distributions are supported on
characters.
As Arthur explains in [38], there are at least two forms of the problem, for all invariant distributions
or for all invariant tempered distributions, neither of which is completely solved, and neither of which isappropriate or necessary for the question at hand. Thus, in [10] a false start was made with Assumption
5.1, an assumption that may be valid but has not yet been established in general. There are, so far as I cansee, two important ways in which his final arguments differ from those foreseen in [10], which remains
none the less a basic reference. Indeed, it appears that to someone with a fundamental understandingof all that is contained in [10], the later modifications are minor, but for those who are attempting
to reconstruct the arguments for themselves in the belief that there is a linear progression from [10]through [26], [27], [28] there are pitfalls, from which I had to be rescued by Arthur himself.
Although we mentioned in passing that it was best to consider the trace formula as applying
to functions on G1 = G(A)1 , the set of elements x in G(A) such that |χ(x)| = 1 for every rational
character χ of G over the base field, we were content to deal implicitly with the case that there were nosuch rational characters, so that G(A) = G(A)1 . We recognized, however, that this assumption was not
valid for the Levi factors, so that it was quite inappropriate as almost all arguments in Arthur proceedIntroduction to the work of James Arthur

31

by induction. Since we have now to explain an important induction argument, the assumption, madeonly for notational purposes, is abandoned. The trace formula becomes a distribution on G(A)1 , but
its form is not at all changed. This is at first surprising, but then one observes that all terms on thegeometric side are concentrated on G(A)1 , because |χ(γ)| = 1 for every γ ∈ G(F ), and that all termson the spectral side will continue to be concentrated on G(A)1 because they will include an integral
over ia∗G , the dual of G(A)1 \G(A). (I made no attempt to describe exactly the terms on the spectralside that result from Theorem 8.2 of [14]. They are contributed by Levi factors L and contain an integral
over ia∗L /ia∗G , which is replaced by an integral over ia∗L when an integration over ia∗G whose effect is acontraction to G(A)1 is added.)
The critical feature of the passage from a family of noninvariant distributions {JL } to a family{I L } was the existence of the distributions I L appearing in formula (36). In lieu of a general theorem,
it would be enough to show that the distributions appearing in the trace formula are supported on
characters. This is accomplished by an elaborate induction in [26], [27] that is one of the new featuresof the argument.
The notion of a distribution being supported on characters depends of course on what are to
be considered the pertinent distributions, those on functions with compact support, those on rapidly
decreasing functions, or those on some other space. The initial choice, in [10], seems to have been
tempered distributions, but for lack of the necessary results in this case that paper remains incomplete.
So Arthur prefers general distributions, but he then encounters another difficulty. The images φM (f )of the map φM is not necessarily a function of compact support even when f is. It is not easy to find
one’s bearings on this constantly shifting terrain. For the purposes of his arguments, Arthur introduces
spaces of functions that are neither slowly decreasing nor compactly supported and that, as we alreadyobserved, are in addition taken to be K -finite.
Recall that U (G) and V (G), or more generally U (M ) and V (M ), are to be defined with respect toa finite set S of places. The functions in U (G) are not necessarily of compact support on G(FS ), but
their restrictions to every slice

GX = {x ∈ G(FS ) | HG (x) = X}
are. The space V (G) is defined in a similar way. Arthur is forced to some such definition for he needs toshow that if φM (f ) is defined by (37), then there is a function f  in U (M ) such that tr(f  (π)) = φM (π)
for all π .

Introduction to the work of James Arthur

32

Even more care and even more elaborate notation is necessary asf (π) =G(FS )

π(x)f (x)dx

may not be defined. The integral over a slice is, and one can work with integrals over slices, so thata function f in U (G) defines not a function on Πtemp (G(FS )) but a function on Πtemp (G(FS )) × aG
(Arthur’s notation in [28] is more complicated, because he has to envisage a more general possibility.)
If π is a tempered representation of G(FS ) induced from a tempered representation σ of a Levi factor
over FS , and this has to be taken to be a collection of perhaps quite disparate Levi factors, one for eachv ∈ S , then the trace on a slice tr(π(f X )) can be calculated as usual by attaching to f a function fM
on M (FS ) that will be compactly supported on each slice with respect to aG . Each aMv is likely to belarger than aG and we can multiply σ with a character eΛ(HM ) of aM =v aMv to obtain σΛ . SupposeXπΛ is induced from σΛ . Since we can calculate tr(πΛ (f X )) as tr(fM(σΛ )), it will behave modulo aG

like the Fourier transform of a function with compact support. This has to be taken into account whendefining the space V (G) and made part of the definition. Otherwise the map from U (G) to V (G) willcertainly not be surjective. It is, moreover, no longer obvious that φM maps U (G) to V (M ). This isproved in [28] and looks to be a second new feature of the argument, although the difficulties had
already been broached in [10] because Arthur treats there the case that U (G) is a space of compactly
supported functions and V (G) is defined accordingly. He points out to me that there is a difficulty thatwas overlooked in [10] (but not in [28]!). The split component aM of a Levi factor may grow when wepass from a global field to a local field because its dimension is that of the maximal split subgroup of
the centre of M . This means, as we already observed, that the geometry that subtends the constructionsof §2 generally changes and this was not noticed in [10]. The passage from the set S to a single placev and from the global geometry, or rather the intermediate geometry over the finite set S , to the local
geometry is effected by the splitting and descent formulas of §7, §8 and §9 of [26] that are deduced fromproperties of the convex sets appearing in §2 of this report.
The formula (36), in which we now suppose that all the c(M ) are 1, can be applied to bothLL
J L (f ) = JM(γ, f ) and J L (f ) = JM
(π, f ) (M ceasing for a moment to be the variable Levi factor on

the right of (36) and becoming a given Levi factor). If we want to define IL and IˆL inductively, then at
the stage G we first define I G by

I G (f ) = J G (f ) −
M ∈L(M0 )M =GI M (φG
M (f )).

Introduction to the work of James Arthur

33

Provided that all the I M on the right are defined, the distribution IG is well defined and easily shown
to be invariant. The problem is then to show that this particular distribution is supported on charactersso that it can be represented as a distribution IˆG on V (G).
Here as elsewhere, Arthur, in the style of Harish-Chandra, argues relentlessly in general, without
examples. In contrast to those of Harish-Chandra, which are usually algebraic or analytic, Arthur’sarguments are often implicitly geometric, so that they cry out for illustration. This is so for those basedin one way or another on (G, M )-pairs, and is perhaps even more urgent for the arguments with which
he establishes that the distributions IG are supported on characters. There are two parts to the proof.
The second uses the trace formula itself and an idea of Kazhdan to complete the final step which is toG
show by induction that the distributions f → IM(γ, f ) are supported on characters.

I do not pretend to understand the first part of the proof, but Arthur was kind enough to offersome insights that will certainly be useful to anyone studying [26] and [27]. In principle, it is to showG
G
that if the distributions IM(γ) are supported on characters then so are the distributions IM(π). In fact,Gthe distributions f → IM
(π, f ) do not appear alone in the trace formula; they appear only as in (14)G
in integrated form. It is an integrated form of IM(π) that is shown to be supported on characters. It
G
is a direct formal consequence of the formula (36) that implicitly defines IM(π) in terms of the variousLGJM(π) that IM(π) is 0 if π is a tempered representation of M (A). There are, however, nontemperedG
representations that will occur in the trace formula. For these, IM(π) may not be 0, but it is only an

integrated form that appears and the distribution does vanish for π tempered. If the rank of M in G
is one, these would be the representations on the imaginary axis, so that – in the simplest case that

π is induced from σΛ with Λ not purely imaginary – the integral can be deformed leaving only theGresidues. These residues appear in the Fourier transform of IM(π) as exponentially decaying terms
GG
that are not cancelled by anything in JM(γ). Thus they must appear in IM(γ), so that the integratedGGIM(π) is supported on characters if the distributions IM(γ) are.

Thanks to the treatment of the geometric side in [21], the left side of (40) can be readily converted
into the form (14) ([27] §3). For the spectral side, the basic formula is that of Theorem 8.2 of [14] forwhich I have referred to the original paper. The factor aM (π) that appears in (14) is defined by thescalar normalising factor in the global intertwining operators and thus can be expressed in terms oflogarithmic derivatives of automorphic L-functions. Although in the formula (14), taken from [26],there appear to be no problems of convergence, this is not so. This is emphasized by the less elegant
form of the formula that is given in [27]. It is to be hoped that the results of [M] will some day be
improved to yield unconditional convergence.

Introduction to the work of James Arthur

34

7. First applications. An application of the trace formula that shows its importance as a primary
analytic tool in the theory of automorphic forms is to the proof that the Tamagawa number of a simplyconnected semisimple group is 1. This is a modern, general formulation of results of Eisenstein, Smith
and Minkowski on class numbers of quadratic forms. There are two steps in the proof. For quasi-split
groups it is an immediate consequence of the elements of the theory of Eisenstein series. For a general
simply-connected semisimple group G, it is obtained by comparing the trace formula for G with thatfor its quasi-split form ([K1]). Although some of the results from [10], [21] and [24] are required, theproof, like that for quasi-split groups, is to be regarded as belonging to the very elements of the analytic
theory of automorphic forms. There were other difficulties, arising from instability or endoscopy, toovercome, but Kottwitz was able to finesse these.
Although this is not clear from the publication dates, the first application of the general trace
formula was, however, to base change for GL(n). As in the proof that the Tamagawa number is 1, it is aquestion of comparing two trace formulas, but now one is a twisted formula, for the group GL(n) overa cyclic extension E of the global base field F and its outer automorphism A → σ(A), σ a generatorof the Galois group of E/F . The value of twisted trace formulas was first recognized in a particularcase by H. Saito ([S]). The truncations required to imitate [7] and [9] were, I believe, first found, but not
published, by Kottwitz. Because twisted trace formulas are so important for applications, in particular
to the applications that appear to be his goal, Arthur has developed most of his later arguments in thisbroader context. Having enough to explain already, I preferred not to introduce this extension in theprevious discussion and shall, by and large, leave it to the reader to consult either Arthur or his own
imagination for a formulation of the necessary results and techniques.
When E/F is of prime degree, the trace formula compares the representation of G = G(AF )on L2 (G(F )\G(AF )) with the natural action of the semidirect product G = G(AE )  σ , a subset ofG(AE )  Gal(E/F ). There is a norm map from conjugacy classes in G to conjugacy classes in G
that, as recognized by Shintani, links the harmonic analysis on the two groups, or more precisely theinvariant harmonic analysis on G and the twisted invariant harmonic analysis on the set G. Associatedto the norm map is a transfer8 from functions f on G to functions f  on G . It is first defined locally,on the corresponding objects over local fields, and then, by taking products, globally. Given f thefunction f  is well defined for the purposes of invariant harmonic analysis, but not uniquely defined.
Basically, orbital integrals of f  are equal to twisted orbital integrals of f . This permits a comparison of
8 The role of primed and unprimed objects is at ﬁrst glance disturbing. The guiding principal of [30], from which
the notation is taken, is that the group over E is the primary object and the group over F only one of its endoscopic
groups.

Introduction to the work of James Arthur

35

the twisted trace formula for f with the ordinary trace formula for f , from which the results on basechange of [30] are obtained.9
A second, related comparison is between the multiplicative group G of a simple algebra over F
and its quasi-split form G = GL(n) that in principle yields extensions of the results obtained in [JL]for quaternion algebras. Since the two groups G and G are now isomorphic at almost every place, theconstruction of f  from f is easier.
In both applications, as in many others, the starting point is the observation that the nonvanishingprincipal elliptic terms (those given by regular semisimple classes o on which the orbital integrals off  or f do not vanish) for the two trace formulas to be compared are in bijective correspondence and,because of the construction of f  from f , pairwise equal. It has then to be deduced from this, fromthe two trace formulas, and from general features of spectral analysis that the discrete parts of the two
spectral sides are then equal. The conclusion is either base change from F to E for the group GL(n) ortransfer of automorphic representations in the sense of functoriality, as in [56], from the multiplicative
group of a simple algebra to its inner form GL(n).
The obstruction to an immediate inference are the remaining terms, on both the geometric andthe spectral sides. There are four expressions appearing in the formula (14), or its twisted analogue,
whose nature and meaning are obscure: aM (S, γ); aM (π); IM (γ, f ); and IM (π, f ). There will besimilar expressions occuring in the trace formula for G . It is, in fact, best to focus on the twisted case,thus on base change, in which more complications appear. Thanks to Hilbert’s Theorem 90, whichcharacterizes elements with the same norm, the geometric terms in the two trace formulas can be put
in bijective correspondence. This can also be done for the spectral terms if some of those for G are firstcollected together in a way that is straightforward in principle, but subtle in fact. It is then proved,using induction and the trace formulas, that each of the four expressions for G, and for γ or π as thecase may be, is equal to the corresponding expression for G and γ or π .
To collect terms, the local base change or lift10 is first defined for tempered representations using
the striking fact observed by Shintani for the group GL(2) (and for some groups over finite fields as
9There is a very diﬃcult, and in large part unresolved, local problem implicit in the passage from f  to f , thefundamental lemma. It is a problem to whose partial solution many mathematicians have contributed, and the trace
formula can usually only be applied after the pertinent form of it is available. For the purposes of [30], a special but
important case treated by Kottwitz suﬃced.
10 I observe, for the sake of those to whom this use of the word lift suggests a retreat on my part, that, on thecontrary, this is the sole context in which it seems an appropriate substitute for transfer. Geometrically a Galois
extension corresponds to a covering and the transfer to a lift from the base space (the scheme deﬁned by F ) to thecovering space (the scheme deﬁned by E). Since the word transfer is now used in quite a diﬀerent sense in endoscopy,
there is good reason to search for a new term for functorial transfer, not otherwise taken, that, in contrast to lift,
does violence neither to our mathematical intuition nor to our linguistic sensibilities (as in the pleonasm base-changelift) and that, in contrast to transfer, has a chance of gaining general acceptance.

Introduction to the work of James Arthur

36well) that, apart perhaps from a sign, the twisted character of a tempered representation at an element

γ in G(Ev ) is equal to the value of the character of the representation from which it is obtained by basechange at the norm γ . The argument is however global: the local existence is deduced from the trace
formula itself, applied as suggested by Deligne and Kazhdan to a class of functions carefully chosen
at two places – but otherwise arbitrary – so that, thanks to splitting formulas, all the difficult terms
become zero. Once local base change is available, global base change is also defined although it is
not assured, without further argument, that the base change of an automorphic representation is again
automorphic. In principle, the collection is effected simply by putting together all representations withthe same lift. It is, however, more difficult than that because Shintani’s principle is not valid for many
nontempered representations.
In passing, I observe that the continuing inability to establish the unconditional convergence ofthe right side of (14) entails complications in the proofs of [30].
There are four different types of equalities to establish, one for each of the four expressions. Two,those for aM (S, γ) and aM (π), can be considered as global in nature. In so far as the first of these
are related to Tamagawa numbers, the desired equality is easier, but, as the discussion prior to (32)suggests, the factor aM (S, γ) will also contain a unipotent contribution to which there is no directaccess. The other two are local and are in part amenable to the technique of Deligne-Kazhdan.
Many of the equalities follow from an induction assumption and splitting principles. (I confess that
I have yet to understand this part of the argument in any serious way.) Thus when the two geometricalsides are subtracted, one from the other, there is a good deal of cancellation almost immediately. Thereis also a good deal of cancellation on the spectral side. Indeed, all that is left upon subtraction is thecontribution of the discrete spectra. On the geometric side, what appears is a sum over Levi factors of
the sum over γ  of(41)

IM (γ, f ) − IM (γ  , f  ),the first arising from the ordinary trace formula, the second from the twisted formula.
If it were possible to remove the sum over M , replacing it by a given M = G and if the differencesin (41) were the orbital integrals of a function h on M (A) of the type exploited by Deligne-Kazhdan,thus for which all terms of the geometric side of the trace formula except those corresponding to elliptic
semisimple classes are zero, then we could apply the trace formula to M and h. Comparison of thespectral side of the new trace formula for M and h with the difference of the spectral sides of the
trace formulas for G and G leads to an equality between expansions for, on the one hand, a discreteIntroduction to the work of James Arthur

37

spectrum and, on the other, a continuous spectrum. Both sides of such an equality have to be zero. Thisis, of course, a standard device in applications of the trace formula, but its successful use here requiresthat a large number of difficulties be overcome. In particular, it has to be shown that the singularities
that may occur in the two terms of (41) individually but that may not appear in the difference if it is tobe an orbital integral cancel each other. The upshot is, finally, the other three types of equality having
been established along the way, that the remaining one, that for aM (π), is valid as well.
Both base change and the calculation of the Tamagawa number are the result of comparing thetrace formula for two different groups. There is a second type of application, still in an embryonic stage,
as only the simplest11 of examples have been treated in order to avoid all difficulties arising either fromthe existence of any endoscopic groups but the principal one or from noncompactness ([K2]). Here thetrace formula, applied to a special choice of function, is to be compared with a formula of Lefschetz
type, usually for a correspondence on an incomplete algebraic variety over a finite field, the reductionof a Shimura variety. What can be concluded from [32] (see also [25], [31]) is that the trace formula
yields a result that, with the help of endoscopy, will be comparable12 with the results from the Lefschetz
formula – when they become available.

8. Local harmonic analysis. The local harmonic analysis to which the heading refers is all a consequence of the local trace formula, but this is not immediately apparent from the earlier results.
The two papers [7], [9] on the first trace formula for a general group appeared in 1978 and 1980,
but they were preceded by the paper [1] that dealt with groups of rank one. It was easy to anticipate on
the basis of the formulas in this simpler case some of the weighted orbital integrals that would appear
in general. They are essentially local objects and their analysis, at least at the infinite place where theanalytic techniques developed by Harish-Chandra were available, could be taken up immediately. So
it is not surprising that the earliest results on them ([2], [4]) appear before the general trace formula.
For p-adic groups, the principal theorem of [4] does not appear until much later, in [23], but the proofsof [4] and [23] have much in common, Since the theorem turns out to be an illustration of the local trace
formula, it is useful to look more carefully at it.13
For a compact group, the integral of a matrix coefficient of an irreducible representation overa conjugacy class is, easily identifiable constants aside, equal to the character of the representation
11
12
13

Simple but diﬃcult!
With the help of the fundamental lemma!
Arthur has observed to me that the results of [23] are exploited in the proof that singularities cancel in thecomparison of the trace formula for GL(n) with that for the multiplicative group of a simple algebra (see p. 119 of[30]).

Introduction to the work of James Arthur

38

evaluated on the conjugacy class. It was discovered in the fifties that a similar formula is valid forsquare-integrable representations of any reductive group over R, but only for elliptic conjugacy classes,
thus the classes of those regular elements that are contained in a Cartan subgroup compact modulo thecentre. The orbital integrals over the other classes is zero (the Selberg principle). It is shown in [4] thatfor each of the other Cartan subgroups T there is a similar formula, but the orbital integral must be
replaced by the weighted orbital integral defined by the Levi factor M for which AM is the maximal
split torus in T . Moreover there is a sign that appears. Although the character of a square-integrable
representation is not given by a simple formula off the elliptic elements, there are clear combinatorial
procedures for computing it, so that, as a result of the formula of [4], the weighted orbital integralsof the matrix coefficients of square-integrable representations can be, in principle, calculated. It is theformula of [4] that allows Arthur to deduce from the trace formula a formula for the trace of the Hecke
operators that it will be possible to compare with the Lefschetz formula.
However difficult it is to understand all the details of [4], the basic ideas are readily accessible
to an older generation familiar with the techniques of Harish-Chandra. There is an induction on thedimension of AM , starting from AG which is of dimension zero and for which the theorem is that ofHarish-Chandra for ordinary orbital integrals of matrix coefficients. It is a matter of using the actionof the centre of the universal enveloping algebra to establish the appropriate differential equations
and the appropriate boundary conditions at the points where two tori, one of which has one compact
dimension more than the other, meet.
As observed, these formulas can be deduced from a general formula, the local trace formula, which
resembles the trace formula and is proved in a similar way using the truncation of a kernel associated
now to a function f on G(F ) × G(F ), F being a local field. The formula, whose possibility wassuggested by Kazhdan, is proved in [39] and is used in a number of ways to overcome difficulties thatarise in applications of the trace formula. Suppose thatf (y1 , y2 ) = f1 (y1 )f2 (y2 )is a smooth, compactly supported function on G(F ) × G(F ). The left and the right actions or G(F ) on
itself yield an action of the product on L2 (G(F )) and thus an operator associated with f , ϕ → ϕ ,

ϕ (x) =

G(F )


G(F )

whose kernel is

f1 (u)f2 (y)ϕ(u−1xy)dudy,K(x, y) =G(F )f1 (xu)f2 (uy)du.

Introduction to the work of James Arthur

39

The local trace formula is obtained by restricting this kernel to the diagonal, truncating, andintegrating. In appearance it resembles (14). On both sides there is a sum over Levi factors containing
a given minimal one.



WM
DM/G 0G
W0
M





WM
JM (γ, f )dγ =DM/G 0G
W0Γell (M )MΠdisc (M )aMdisc (π)JM (π, f )dπ,

where for the sake of a compact formula I have set (−1)dim(AM /AG ) = DM/G . More explicitly, thegeometric terms are given byJM (γ, f ) = D(γ)

−1f1 (x−1
1 γx1 )f1 (x2 γx2 )vM (x)dx1 dx2 ,AM (F )\G(F ) AM (F )\G(F )where the weight vM (x) = vM (x1 , x2 ), x = (x1 , x2 ), is again defined by a (G, M )-family,vP (Λ, x) = e−Λ(HP (x2 ))+Λ(HP̄ (x1 )) .
The set Γell (M ) is the set of conjugacy classes (γ, γ), where γ is elliptic in M . Thus the maximal split
torus in the centre of its centralizer is AM .
The spectral side of the local trace formula, like the spectral side of the trace formula itself,
is deceptive because there is considerable complexity veiled by the simple notation Πdisc (M ) andJM (π, f ). The representation π is a representation of the product G(F ) × G(F ). To free the symbol
M , we define Πdisc (G) for an arbitrary G, which can then be one of the Levi factors M . The collection
Πdisc (G) is formed from constituents of induced representations
IP (σ̌ ⊗ σ) = IP (σ̌) ⊗ IP (σ),
in which σ , with contragredient σ̌ is a square-integrable representation of M (F ), M the Levi factor of

P , invariant under an element t of the normalizer of AM such that adt − 1 is invertible on aM /aG .
The parabolic subgroup P is, as usual, taken to be defined over F and to contain a fixed minimal one.
The factor adisc (M ) is simpler than the corresponding global factor and involves no transcendental
elements, but there is little to be gained by describing it in full. The terms JM (π, f ) are, like the global
factors, associated to (G, M )-families defined by intertwining operators, now local and unnormalized.
Although the local trace formula, as it first appears and as it is presented above, is not invariant, it,like the global formula, has an invariant form that is given in [36] and, in more detail although still at
a somewhat breathtaking pace, in [40]. The formal appearance of the geometric side does not changeIntroduction to the work of James Arthur

40

much on passing to the invariant formula, although there is an important splitting formula14 for theL(γ, fi,Q ), i = 1, 2, definedinvariant distributions IM (γ, f ) that permits their separation into those IM

by weighted orbital integrals for the group G itself. The spectral side of the invariant formula has an
elegant form,



(42)

WM
DM/G 0G
W0
iM (τ )rM (τ, f1 × f2 )dτ.
Tdisc

The integration is over an elegantly chosen collection of virtual representations and iM (τ ) is an elementary factor. What is striking, although on reflection perhaps not surprising, is that in the termrM (τ, f1 × f2 ) = rM (τ, P )Θ(τ̌, f1,P )Θ(τ, f2,P ),(43)

the contributions from f1 and f2 are separated into invariant distributions defined once again byvirtual characters. I omit their explicit description; it is related to the decomposition of tempered
induced representations into irreducible representations. The factor rM (τ, P ) is defined as usual by a(G, M )-pair, one attached to local scalar normalising factors. So the intertwining operators themselvesappear on the spectral side of the invariant local trace formula only implicitly, through the virtual
characters τ .
The earlier formulas ([4], [23]) for weighted orbital integrals as characters can be quickly deduced
from the invariant local trace formula. Indeed they play, curiously enough, a role in its inductive proof.
Although, as reported above, Arthur had earlier established, in the course of proving the existence ofan invariant global trace formula, that the distributions IM (γ) are supported on characters, the local
trace formula allows him to prove this by purely local methods that are, as before, inductive.
The new proof of the earlier formulas is sketched in [36]. A more detailed proof of a more generalformula appears in [40]. Suppose M is a given Levi factor. Take f1 to be a pseudocoefficient (almost thesame thing as a matrix coefficient) of a discrete-series representation π and take f2 to be such that its
orbital integral is an approximation to a δ -function at a given element of Γell (M ). The splitting formulafor IM (γ, f ) reduces the geometric side to the term involving an integral of IM (γ, f1 )IG (γ, f2 ). Taking
the limit in f2 leaves IM (γ, f1 ) at the given element γ which is the same as JM (γ, f1 ) because f1 is a
pseudocoefficient. On the other hand, Θ(τ̌, f1 ) will be zero unless τ is π and Θ(π, f2 ) will be given by
the value of the character of π at the given γ .
14 It could perhaps just as well be called a descent formula. This report is, in spite of its many pages, far too brief
and too hastily written. There is much material that, with more time and more space, I would have liked to treat
at greater length. There are many things, among them splitting and descent, over which we might have lingeredwith both pleasure and proﬁt. Arthur himself does not always ﬁnd time to do his own ideas justice.

Introduction to the work of James Arthur

41

It has already been observed that the formula of [23] was used in [30] in the course of establishing
the cancellation of singularities. A number of other consequences of the local trace formula are, orwill be, invoked in later papers in which the trace formula is applied to functoriality. Orthogonality
relations for characters are familiar from the theory of finite or compact groups, and even, following
Harish-Chandra, for the discrete-series representations of reductive groups. There is an extensionof the orthogonality relations of Harish-Chandra to the class of elliptic representations, which are inessence, the tempered representations whose character does not vanish on the elliptic elements. It is
indispensable in comparisons of trace formulas. In abbreviated form, as it appears in [40], it is


(44)


Γell (G(F ))Φ(γ)Φ (γ) =Tell (G)φ(τ )φ (τ )dτ.

The left side is in essence the integral of the product of two characters over the elliptic set, thus exactly
what appears in the formula of Harish-Chandra; the right side is a sum not over elliptic representations
themselves, but over a more convenient set of virtual representations the span of whose characters is
also the span of the characters of the elliptic representations.

9. Unipotent representations. Although functoriality (see, for example, [56]) as originally enunciatedsuggested many new problems in the theory of automorphic forms and provided a coherent way ofviewing otherwise disparate phenomena, there was behaviour that did not obviously fall within its
scope until Arthur began to reflect on the prerequisites to developing the stable trace formula beyond
its embryonic stages ([KL]) and to comparing trace formulas for different groups. He was led to aseries of elegant conjectures ([18], [34], [35]) that are rich in consequences, some of them proved ([ABV],[MW,Mö]), and that throw a great deal of light on outstanding problems in both the global theory andthe local theory.
Recall that basic to the notions of functoriality is the possibility of attaching to a connected reductive
group G over a global or a local field F a complex group, its L-group L G, defined in essence by passing
to the complex group15 whose Cartan matrix is the transpose of that of G. When F is global, it is thenpossible to attach, by means of the Hecke operators, to any automorphic representation π a sequence{Ap } of conjugacy classes in L G, aptly called Hecke classes or with the case that G = {1} in mind
Frobenius-Hecke classes, and defined for almost all places p of F .
If G = GL(2) then L G can be taken to be GL(2, C). Then the classes are described by the two
eigenvalues αp and βp of Ap . The Ramanujan conjecture, its generalization, the Ramanujan-Petersson
15 It is an extension of this connected group by a Galois group or by a Weil group of a suﬃciently large extension
K of F , the choice between the two possibilities and the choice of extension being dictated by circumstances.

Introduction to the work of James Arthur

42

conjecture, and the generalization of that to Maaß forms and to arbitrary number fields assert that

|αp | = |βp | = 1 for all p (at least for all p for which the class is defined) if π occurs in the space ofcusp forms. The complete conjecture asserts that the local components πv are tempered at all places,
including those at infinity. The generalization of this to arbitrary groups would, at the first, carelessglance, suggest that the class {Ap } met the maximal compact subgroup of L G if π occurs in the spaceof cusp forms or, when the archimedean places are included, that πv is then tempered for all v . In thisform, the conjecture was shown very early on to be invalid ([HPS]).
Arthur’s conjectures suggest that it is invalid because the notion of cusp form, useful as it is forthe spectral theory, has no structural significance. There is no reason, beyond ingrained scepticism, todoubt that the generalized Ramanujan conjecture is valid for GL(n) and Arthur’s global conjectures
and their meaning are best discussed in the firm conviction of its truth in this case. The noncuspidal
discrete spectrum for GL(n) does not satisfy Ramanujan’s conjecture and was never expected to. It
was described by Moeglin-Waldspurger, who verified thereby a conjecture of Jacquet. Rather thanpresenting the explicit descriptions of the representations that occur, I describe the Arthur parametersthat arise naturally from the constructions of the representations and provide the right optic for thegeneral conjectures.
Among specialists there is a fairly widespread belief that the (unitary) irreducible cuspidal automorphic representations of GL(n) will be classified by the (bounded) n-dimensional irreducible
representations of a group LF . This group, if it exists, will be unconscionably large, the inverse limitof Lie groups that are almost compact (or sometimes of their complexifications), and presumably inno real sense explicitly describable. The mere fact of its existence would, however, be equivalent tofundamental properties of automorphic forms, so that it is well to keep this hypothetical group in mind
and to make every effort to render it – or at first the consequences of its existence – concrete. As aresult of the conjecture of Jacquet proved by Moeglin-Waldspurger, the discrete automorphic spectrumof GL(n) is parametrized by pairs (σ, m), where m divides n and σ is a cuspidal automorphic representation of the group GL(m). Thus σ is attached to an m-dimensional representation φ of LF . Arthur
suggests that we complete LF by multiplying it by the Lie group SU (2), take l = n/m and take as theparameter of the representation π defined by (σ, m) the tensor productψ = ψss ⊗ ψunip = φ ⊗ ψunip ,where ψunip is the unique irreducible l-dimensional representation of SU (2) and ψss = φ. The
parameter ψ is referred to as the Arthur parameter of π . This formulation is to be justified by what itIntroduction to the work of James Arthur

43

predicts for automorphic representations on groups other than GL(n) and, also, for what it predicts
about the local theory. It turns out that it suggests a great deal, some of which can be proved.16
If π is attached to the parameter ψ then the sequence {Ap (π)} is given by
Ap (π) = Ap (σ) ⊗ ψunip

Nmp1/200
Nmp−1/2



So it is clear that π does not satisfy the Ramanujan conjecture when l > 1. The simplest examples arefor m = 1. Then π is a one-dimensional representation.
For local fields F , especially but not alone the real and complex fields, the group LF , whose
irreducible representations of dimension n now classify the tempered discrete-series representationsof GL(n, F ), is simply described. It is the Weil group WF of F if F is real or complex and the product

WF × SU (2), a variant of the Weil-Deligne group, if F is nonarchimedean. The tempered, unitary
representations of GL(n, F ) are classified by the bounded representations of LF , irreducible or not,of dimension n and arbitrary irreducible representations of GL(n, F ), unitary or not, by arbitraryn-dimensional representations of LF .
A similar classification has been established for all reductive groups over archimedean fields,
representations of dimension n being replaced by homomorphisms

φ : LF → L G.
There is every reason to expect that such a classification is valid over all local fields, but the questions
elucidated by Arthur’s conjectures arise already for the real field. It is not a single irreducible representation of G(F ) that is attached to φ, but a finite set, an L-packet, of inequivalent representations
that have the same L-functions, so that as arithmetic objects they are hardly distinguishable. If the

L-packet consists of tempered representations its internal structure is described by the theory of endoscopy, established for real groups by Shelstad. Otherwise this theory, which amounts to a collectionof identities between linear combinations of characters of G(F ) and characters of lower dimensionalgroups, may fail. The local form of Arthur’s conjectures reestablish the theory not for all representations but for a very important class that includes presumably all local factors of all unitary automorphic
representations.
16 We shall be concerned in the rest of the report with the researches of Arthur himself, which are, above all,consecrated to the consequences of this point of view for the trace formula and for functoriality. For consequences
pertaining to the very diﬃcult problem of classifying unitary representations of reductive groups over the real ﬁeld
and other local ﬁelds, see, for example, [ABV]; for those for the classiﬁcation of the discrete automorphic spectrum,see [Mö].

Introduction to the work of James Arthur

44

These representations are to be parametrized by maps,(45)

ψ = φ × ψunip : AF = LF × SU (2) → L G,

so that the classification of the tempered packet is recovered from the ψ for which ψunip is trivial. The
existence, even for the real field, of the unitary representations needed for the Arthur packet Πψ is by
no means evident and, in general, not yet known.
The factor SU (2) that appears in (45) is for nonarchimedean fields a second factor of this type.
For both archimedean and nonarchimedean fields, and globally, the SU (2) of (45) seems to havealgebro-geometric content, a part of which, its relation to multiplication by the fundamental class in
Hodge-Lefschetz theory, is described in §9 of [34]. For the trace formula, and in particular for global
endoscopy, it is the character relations for Πψ that are critical. The pertinent object here is the group Sψ
of connected components of the centralizer Sψ in the connected component of the identity in L G of theimage ψ(LF ). The character identities are defined by a pairing between the L-packet Πψ and Sψ . Eachs in Sψ defines an endoscopic group H , also reductive and of dimension no larger than that of G itself.
There is (almost) an imbedding of L H in L G and ψ factors through L H , so that it defines a packet for

H . If χπ is the character of π and s ∈ Sψ thend(s, π)χππ∈Πψ

is simply expressible in terms of the stable character associated to the Arthur packet for H associatedto ψ. Moreover, for each π the function s → d(s, π) is essentially a character of Sψ
There is, of course, a troubling vagueness to these statements that is not present in [34], which,however, not only assumed a familiarity with the aims of endoscopy and its earlier results but also someacquaintance with the available unipotent representations (those associated to parameters ψ for whichψss is trivial). Since endoscopy for tempered representations, and all the more for Arthur packets,is attempting to express concisely a great deal of information of which only fragments are presently
available, fragments that are generated by difficult theories that draw on quite diverse sources fortheir methods, the choice is between vagueness or a careful treatment of many examples. Rather thandescribing local examples, I pass, however, to the global conjecture and to its implications.
In contrast to the local conjecture, which does not provide a parametrization of all unitary representations, the global conjecture is thought to provide a description, in terms of LF , and thus interms of representations for which the Ramanujan conjecture is satisfied, of the complete spectrum ofIntroduction to the work of James Arthur

45

L2 (G(Q)\G(A)). Unfortunately, however, and again in contrast to the local theory, the group LF is
not given externally to the theory, but has to be constructed within it, perhaps with the help of trace
formulas that build on those of Arthur. It is not yet available, so that Arthur in his attempt to establish some quite general cases of functoriality ([45]) is forced to employ constructions that are strongly
influenced by the possible existence of LF but in which LF is not allowed to appear explicitly.
This said, the global conjecture still envisages a parametrization by the maps (45), now global. The
packets are usually infinite, being defined by the local packets and the relation between the local andglobal LF . The global Sψ has an even more subtle definition than the local and has a slightly different
purpose. The global Arthur packets are products of the local. Each element of each local packet defines
a function on Sψ . Multiplying them together, as is likely to be possible, we can attach to each element

π of the global packet a function on Sψ that determines (conjecturally) whether π occurs globally
and, more precisely, whether it occurs in the discrete spectrum globally and with what multiplicity.
There are subtle aspects to the pairing between π and s ∈ Sψ that determines this multiplicity, mψ (π).
In particular there is an important sign factor that was introduced to respond to the exigencies ofcomparing trace formulas and that is easy, but dangerous, to overlook ([R]). Arthur does not suggestthat mψ (π) > 0 for at most one ψ, on the contrary.
In the trace formula, the geometric side is to be regarded as the known side, the spectral side as
the unknown. More precisely, it is usually the discrete part Idisc (f ) that is of primary concern in any
comparison.17 Recall that the spectral side is a sum of multidimensional integrals over automorphic
representations. The discrete part of the spectral side is the partial sum over those terms for which the
integrals are of dimension zero. It contains not alone the discrete spectrum in L2 (G(Q)\G(A)) but also
what Arthur refers to as surviving remnants of Eisenstein series, resulting from passing to the limit in
the sin(τ T )/τ -integrals of §3.
We have not and shall not describe the notion of endoscopy, which has to be introduced forcomparison of geometric sides, thus basically for circumventing the discrepancy between, on the one
hand, conjugacy classes in G(F ) over a given ground field F , local or global, which is the pertinent
notion for harmonic analysis, and, on the other, conjugacy classes in G(F ) but with respect to G(F̄ ),which are all that can be compared when dealing, for example, with the same group over two different
fields, one an extension of the other, or two groups that differ by an inner twisting.
17

I suppress any complications in the notation arising from any possible, although unlikely, failure of absoluteconvergence on the spectral side. Moreover, I avoid treating twisted trace formulas explicitly although they are
indispensable for Arthur’s purposes. For the basic notions, the reader should consult [KS].

Introduction to the work of James Arthur

46

The effective use of this notion requires the fundamental lemma, or rather in any given case, acollection of fundamental lemmas, one for each endoscopic group. The lemma available, or assumed,
it is possible to begin to stabilize the geometric side of the trace formula. It is the stabilized geometricsides of trace formulas that are compared; it is the stabilized spectral sides that yield information about
the automorphic spectrum. According to [35], the stabilization of the discrete part of the trace formula
should have the form,(46)

Idisc (f ) =H
ι(G, H)Sdisc(f H ).

H

The sum is over the endoscopic groups, of which all have dimension less than that of G itself in thetwisted trace formula and for the usual formula all but one. The function fH is a transfer of f fromG to H , similar to the transfer that appears in base change, a particular case of endoscopy. The global
H
distribution Sdisc, for which my notation, sacrificing precision to simplicity, departs from that of Arthur(itself uncertain), is defined intrinsically on H (or at worst on a closely related group), with no reference
to G.
The validity of (46) is by no means evident, even when the geometric side can be fully treated;
the purpose of [35] is to demonstrate that it is a consequence, neither easy nor immediate, of theconjectures18 of [34], or, to put matters in a different light, that the global conjectures of [34], in
particular the role of the sign character in the multiplicity formula, is a consequence, neither easy nor
immediate, of the validity of (46). For someone with a stake in the outcome and a familiarity with
examples, these arguments are persuasive, but, riddled with conjecture as they are, they lead, at firstglance, to nothing definitive, and may even provoke some scepticism in outsiders. In the concluding
pages of [35], Arthur addresses the problem of deducing definitive, fully established results fromthe intuitively and logically appealing, although incomplete, notions of [34] and [35]. The bulk of his
investigations of the succeeding decade – the papers on the local trace formula and those on endoscopy,of which many have appeared ([44], [45], [46], [47], [48], [49], [50], [51], [53], [54], [55]) while others arestill in preparation – are devoted to realising the program for the classical groups sketched there.20
18 It is striking that conjectures for twisted multiplicities are used even to demonstrate (46) for the ordinary trace
formula.
20 He continues, however, to assume the fundamental lemma, a combinatorial problem that has turned out tobe surprisingly diﬃcult. Partial results on the fundamental lemma are available for groups of low rank, so that we
can expect a complete form Arthur’s results to be available soon for the classical groups SO(4) and SO(5) a case of
considerable interest, especially in regard to the multiplicity formula, because SO(5) and Sp(4) are isogenous andSO(5) deﬁnes some of the ﬁrst Shimura varieties after the classical quotients of the upper-half plane. No-one yetknows for certain how the lemma is to be attacked in general, although a combination of methods from topology andalgebraic geometry seems the most likely possibility. There are three forms of the fundamental lemma: for orbital
integrals on the group; for orbital integrals on the Lie algebra; and for weighted orbital integrals. Arthur assumes
them all!

Introduction to the work of James Arthur

47

10. Classical groups and endoscopy. For the group GL(n) neither stability nor multiplicity is aproblem: stability because conjugation in GL(n, F ) itself and conjugation in GL(n, F ) with respect
to GL(n, F̄ ) are the same; multiplicity because Shalika has shown many years ago that cuspidal
automorphic representations occur in L2 (GL(n, F )\GL(n, A)) with multiplicity one. On the other
hand functoriality predicts a close relation between automorphic representations of the classical groups.
For many purposes the L-group of GL(n) can be taken to be GL(n, C); the L group of the symplectic
group Sp(n) to be SO(n + 1, C), that of SO(n + 1), n even, to be Sp(n, C) and that of the split
orthogonal group SO(n), n even, to be SO(n, C). A nonsplit orthogonal group in an even numberof variables n splits over a quadratic extension E of the base field and its L-group is O(n, C), to beconsidered a semidirect product of SO(n, C) with Gal(E/F ). Thus there is a canonical imbedding φof the L-group of each and every classical group into that of a general linear group.
Functoriality predicts that if there is a homomorphism of L-groups

φ : L H → L G,

(47)

then to each automorphic representation π of H there is associated an automorphic representation Π of

G such that the sequence {Ap (Π)} is, at least for almost all p, equal to φ(Ap (π))}. Thus, in particular,
it predicts that to each automorphic representation of a classical group, and in particular to each one
occuring discretely in the L2 -spectrum of the group, there is associated an automorphic representationof a general linear group.
There is a simple and important outer automorphism of GL(n),

(48)
with

θ : A → J −1tA−1 J,



J =


0

1
.
.
.

1




.


0

The final key observation is that the imbedded classical groups are all among the endoscopic groups
for the corresponding twisted trace formula and that, in addition, all twisted endoscopic groups areeither these imbedded classical groups or products of lower-dimensional classical groups. So, Arthur
argues in [35], the possibility exists of using the twisted trace formula for θ to establish functoriality forthe maps (47) and, in addition, to establish the multiplicity conjectures of [34] for the classical groups.
The project that this suggests will, when completed, establish, once and for all, the overwhelmingIntroduction to the work of James Arthur

48

importance of the trace formula for the theory of automorphic forms, but it will not be, I hope, theend. Endoscopy and the trace formula will not be exhausted until, for example, the results of [30] areavailable for all groups. In addition, as I have maintained on more than occasion, but not yet in print, it
is unlikely that the deepest and most fundamental of the problems posed by functoriality will be settled
without extending the trace formula far beyond its present limits; and that will require, no doubt, if nota mastery then certainly a thorough understanding of Arthur’s methods and, in my present estimation,a willingness to combine them with methods taken from a more traditional analytic number theory.
As observed, endoscopy and stabilization are entailed by local considerations. Endoscopic groups
were first introduced to understand in detail how the characters of irreducible representations of thelocal groups G(F ), F a local field, failed to be functions on stable conjugacy classes. (In essence a stableclass is the intersection of a conjugacy class in G(F̄ ) with G(F ).) The discrepancy is described by stablefunctions or distributions (functions or distributions constant on stable classes) on endoscopic groups.
In principle, a given endoscopic group appears in the description for a given representation π if theputative parameter of ψ factors through an imbedding, or something near an imbedding, of L H in L G.
All endoscopic groups are, by their very definition, quasi-split!
A fully developed global theory of endoscopy requires, first, that it be possible to define a stabletrace S = SH for quasi-split groups and, secondly, that it be possible to express the ordinary trace, thus
one side or the other of the trace formula, as a sum20 over endoscopic groups of stable traces,(49)

I(f ) = I G (f ) =ι(G, H)S H (f H ),H

f H being the transfer of f on G(A) to H(A). The ordinary trace is a trace only in an approximate sense;
it is the value of either side of the equation (14). The stable trace is defined intrinsically on H . For the
nontwisted trace formula, there is a unique principal endoscopic group, the quasi-split inner form of

G. For the twisted trace formula, there may be more than one endoscopic group that is in some sense
principal, even an infinite number of them.
The primitive calculations ([KL]) on which the notion of endoscopy was based were for the simplest
part of the geometric side of (14), for the terms for which M = G and γ is semisimple. There are noother terms if G is anisotropic! These calculations almost lead to an equality of the form (49), but there
20 According to Arthur ([53]), only elliptic endoscopic groups are needed, thus those for which the maximal split
component of the centre is that of G itself. This is surprising because the spectral side of [14] contains what Arthur
refers to as surviving remnants of Eisenstein series, but a converzation with him suggests that what might be treatedas a contribution from a nonelliptic endoscopic group can also be incorporated as a part of the contribution from alarger elliptic endoscopic group and that there are sound reasons for doing so.

Introduction to the work of James Arthur49

are supplementary terms that have to be included for a complete matching. The group H may, forexample, be abelian, so that there is no difference between I H and S H . This occurs for the group SL(2)as in [LL], where the geometric expansion of S H for an abelian H , a one-dimensional torus, contains
not only terms f H (γ), γ not central in G, which are matched by combinations of terms of the geometricside of (14) for which M = G and γ is semisimple but also terms f H (γ), γ central in G, which arematched by combinations of terms of the geometric side of (14) for which M = G but for which γ isunipotent and not semisimple.
So the early calculations are far from adequate; a great deal remains to be done. If G itself is aquasi-split group and the trace is the nontwisted trace, then G appears also on the right side of (49),which can then be regarded as defining inductively SG , because f G = f and ι(G, G) = 1. Otherwise,
whether or not the trace is twisted, all terms in the relation (49) are defined and it becomes a relation to
be proved.
The proof that the stable distributions SG exist and that (49) is satisfied is awe inspiring. Thereis no question of doing justice to it here. Since (49) entails a comparison between the trace formulasfor quite different groups, more attention has now to be paid to the exact definition of the normalising
factors that were used to define the distributions IM (γ) and IM (π) and the coefficients aM (π) thatappear in (14). Arthur prefers a new choice whose relation to the old, from which it does not differ
greatly, is described in [47]. He also, before undertaking the proof of (49), modifies in various minor
ways at the beginning of [53] the formulation of the trace formula. The result looks quite like (14), but
the meaning of the terms has changed and the inner summations are over different sets. There are
significant consequences of these apparently minor changes. For example, the suppression of the finite
set S of places from the notation is shown, all changes made, to be legitimate.
Like I G the stable distribution SH will have two expansions, a geometric expansion and a spectral
expansion. Moreover, the relation (49) will be a consequence of term-by-term equalities either on thegeometric sides or on the spectral sides. Arthur is obliged to treat them while labouring under atremendous handicap: neither the local parametrization nor an adequate local theory of endoscopy are
yet available. Some substitute, factitious or in appearance factitious, has to be found for the first, andthe second has to be established in the course of the treatment, with the help of elaborate inductions.21
Although the transfer of functions f on G to functions fH on its endoscopic groups has been
mentioned several times, I have not had the courage to describe it in any detail. The function f H
21 Fortunately, Waldspurger had already established in [W] that the existence of the transfer over nonarchimedeanﬁelds is a consequence of the fundamental lemma for the Lie algebra. Archimedean ﬁelds had been treated by
Shelstad.

Introduction to the work of James Arthur50

is not uniquely defined. Only its stable orbital integrals are and they are to be expressed as linear
combinations of those of f with coefficients, the transfer factors, in whose definition ([LS]), which is notwell understood, a large number of ingredients from the structure theory of semisimple groups andfrom Galois cohomology enter. One of the keys to the success of Arthur’s treatment of endoscopy seems
to be the adjoint relations of [44] and [49] which establish just the right amount of linear independence22
to permit him to express all orbital integrals uniquely as linear combinations of stable orbital integralsof the functions f H .
On the spectral side, an adequate formulation of the results requires the basis of the stable distributions provided by summing over the characters in an L-packet. In the absence of a local parametrization,
except at the archimedean places, the packets are not available. So Arthur is forced ([44]) to define aweak substitute for this basis. The elements of the new basis have no particular significance, exceptthat they are compatible with a natural filtration on the space of orbital integrals of functions in the
Hecke algebra, whose first term is given by the space of functions whose orbital integrals are zero on all
Cartan subgroups but the elliptic ones. To construct the basis he begins with a basis Φ2 (G), essentiallyarbitrary, of the stable elements in the bottom of the filtration, and not just for G but for each of its Levifactors M , and then obtains the full basis by induction of virtual characters from M to G.
The elliptic characters τ ∈ Tell (G) that appear in (44) form a basis of the orbital integrals at thebottom of the filtration, or better, thanks to the pairing of [44], of the bottom of the filtration and of its
dual. If H is an endoscopic group of G and φ an element of Φ2 (H), then f → φ(f H ) lies in this dual,
so that, for each element φ of Φ2 (H), there is an expansion,(50)

φ(f H ) =∆(φ, τ )fG (τ ),Tell (G)

where fG (τ ) = τ (f ) is simply the value of f , supposed to lie in the first term of the filtration.
Arthur proves in [44] that the right side of (50) is a stable distribution φH when G is quasi-split
and H = G. Thus for any G and any of its endoscopic groups H the value φH (f H ) is well defined.23
He shows that, in addition, φ(f H ) = φH (f H ). The argument is similar to those in the proof (49).
Distributions are defined for quasi-split groups and are shown to be stable. They then appear as thestable distributions on one side of a formula that is to be proved and that links the value of a distribution
on a group G at a function f with the value of these stable distributions at the transfers fH .
22 If the local ﬁeld is archimedean, the cohomological properties of G and, consequently, the basic notions appearing in endoscopy are not so simple as those for a nonarchimedean ﬁeld. As a result the necessary independence
fails, but it is reestablished in an elegant way, suggested by Vogan and Kottwitz, by considering several twisted
forms of the group simultaneously ([49]).
23 The analogous result for archimedean ﬁelds can be deduced from the papers of Shelstad.

Introduction to the work of James Arthur

51

There are four different kinds of identity that are proved in order to establish (49); there are local
and global identities for the geometric side and for the spectral side. They correspond to the four
different expressions that appear in (14) – in the original or in the modified form. The terms IM (γ, f )and IM (π, f ) are local because the information they contain refers to only finitely many places;24 theterms aM (γ) and aM (π) are on the other hand global. To formulate the four identities, the constructionof the distributions f → φH (f H ) has to be extended to a product of places. In addition, the construction
has to be applied to the Levi factors M of G. So it is convenient to follow Arthur, as we did in thediscussion of base change, and to denote an endoscopic group of G by G and one of M by M  . Since

we can apply the distributions φM to the function fM obtained by integrating the function f over the
unipotent radical of a parabolic subgroup with M as Levi factor, it can be regarded as a distribution on

G.
The endoscopic group M  will be a Levi factor of one or more endoscopic groups G of G. For agiven M  , Arthur introduces a collection of these G , in which there may be repetitions, that he denotes

EM  (G).
The local geometric theorem is an expression for IM (γ, f ) in the spirit of (49). Although IM (γ, f )is not an orbital integral in the usual sense, we can combine it with the transfer factors associated to an
endoscopic subgroup M  of M to form(51)

IM (δ, f ) =∆M (δ, γ)IM (γ, f ).

γ

If IM (γ, f ) were the orbital integral of a function on M , IM (δ, f ) would be the stable orbital integral
over the stable class δ of its transfer to M  . With the adjoint relations for the transfer factors on M thenumber IM (γ, f ) can be recovered from the collection of IM (δ, f ). The analogue of the relation (49) forIM (δ, f ) is25
(52)

IM (δ, f ) =







G
G
ιM  (G, G )SM).
 (δ, fG ∈EM  (G)GThe factor ιM  (G, G ) is simple and vanishes if G is not elliptic; SM (δ, ·) is a stable distribution on G .

Since the same endoscopic group G may appear more than once in the sum, which is over M  and G ,24

As with the invariant trace formula, the adjective local might be better replaced by semilocal. Indeed there
are two local theorem, one truly local and one semilocal. I shall not discuss the necessary splitting and descent
theorems for the stable terms needed to mediate between them. My capacities are already overtaxed by the attempt
to understand the statement of the four identities.
25 I simplify the notation in order to pass over in silence a number of issues that would otherwise have to betreated explicitly. A particularly important point is that at the archimedean places, endoscopy does not function
within the context of linear combinations of orbital integrals. The transfer of a stable orbital integral from anendoscopic group to G may introduce multiple layers, thus derivatives transverse to orbits.

Introduction to the work of James Arthur52

it would be possible to collect terms and to express the right side as a sum over stable distributions onelliptic endoscopic groups evaluated at f G .
In order to deduce the expansion (49) from (52), it is necessary to relate the factors aM (γ) for G
to those for its endoscopic groups. In the modified trace formula, the sums are no longer over γ in

M (Q) or, more generally, in M (F ) if the global field is taken to be arbitrary. Rather a fixed, finite setof places V is introduced and the sum is over the projection γ on v∈V M (Fv ) of elements γ̇ in M (F ).
There may be several γ̇ that project on a given γ and the new aG (γ) are expressed in terms of the oldaG (S, γ̇), the set S ⊃ V being taken sufficiently large, but otherwise arbitrary. The second ingredientin the stabilization of the geometric side of the trace formula is an identity,aG (γ) =

(53)

 



ι(G, G )bG (δ  )∆G (δ  , γ).

Eell (G) δ

The outer sum is over the elliptic endoscopic groups of G. The inner sum should be ignored; δ is,essentially, the image of γ in the endoscopic group G . The factor ∆G (δ  , γ) is the transfer factor. Whatis new in the formula are the factors bG (δ  ). Defined by the quasi-split group G and the stable classδ  in it alone, they are the stable analogues of aG (γ). The formula (53) is, of course, proved not for justfor G but for all its Levi factors.
The geometric expansion of the stable trace formulas appearing in (49) is then, for any quasi-splitG,
S G (f ) =(54)

 WM 
0M

W0G

GbM (δ)SM(δ, f ),

δthe inner sum on the right being over appropriate stable classes (defined, as in the modified trace
formula itself, by semilocal objects). As Arthur observed to me, if G is not quasi-split, (49) is notentirely a straightforward consequence of substition of the local and global geometric theorems, thus
of formulas (52) and (53), into the geometric side of the trace formula. In (49) the stable trace SG (newnotation) is applied to a function f G , defined by f on G, and in order to prove (49), it has to be shown






G
G
that for such f G and for δ that are not images of classes in G, we have SM) = 0.
 (δ, fArthur does not, however, even for quasi-split groups, simply deduce (54) from (52) and (53).
Rather he proves everything at once, by an inductive procedure, and the consequence of dealing withso many provisional objects simultaneously is a notational and conceptual thicket in which it is very
easy to lose the way.

Introduction to the work of James Arthur

53

The terms on the spectral side of the modified trace formula, like those on the geometric side,
are more cleanly broken into the product of a local term and a global term than those in the original
formula. In particular, the π that appear are unramified outside of the fixed set V , so that outside of V
they are characterized by the associated sequence of Hecke classes{Ap (π) | p ∈/ V }.

(55)

The local (or better semilocal) term, IM (π, f ), in the modified trace formula is associated to a representation



π=

πv .

v∈V

It is understood, moreover that f is now a function on

v∈V G(Fv ), extended when passing to theoriginal trace formula to a function on G(AF ) by multiplying by a product of units in the local Hecke
algebra. As a result, the global coefficient aG (π) that occurs in the modified formula is a sum overthose automorphic representations, π̇ , unramified outside of V with the same local components at theplaces in V as π , but not merely of the original global coefficients aG (π̇). Rather there is an additional
sum over M and, because of the modification of the normalization of the intertwining operators, anadditional factor defined by the sequence (55). A similar observation applies of course to the globalaM (π).
The construction of the local distributions f → φH (f H ) from the equation (50) yields, simply bytaking products, semilocal distributions as well and provides a basis of the span of the characters on

G(FV ) =

26 The analoguev∈V G(Fv ) given by transfers of stable distributions on endoscopic groups.

of (51) is27

(56)

IM (φ, X, f ) =∆(φ, π)IM (π, X, f ),π

a semilocal definition. The analogue of (52) is28

(57)

IM (φ, X, f ) =







G
G
ιM  (G, G )SM).
 (φ, X, f

G ∈EM  (G)26

For the complete theory Arthur refers to preprints that are not yet available. It is necessary to guess at theircontents. In particular, it appears that he replaces the sum of (50) by a sum over π rather than over the virtual
characters τ . Since this is a formal matter, for one is every bit as much of a basis as the other provided we takecare to enlarge the discussion immediately to all characters and not to conﬁne it to elliptic characters, I admit withno compunction factors ∆(φ, π).
27 We have already seen X in §6. It is necessary for technical reasons because I (π, f ) appears in an integrated
M
form in the trace formula.
28 Once again, there are equivalence relations on which I do not insist.

Introduction to the work of James Arthur

54

The analogue of (53) is(58)

aG (π) = ι(G, G )bG (φ )∆G (φ , π).

Eell (G) φFinally, the analogue of (54) is(59)

S G (f ) = WM 0M

W0G

GbM (φ)SM
(φ, f ),

φThus (54) and (59) lead to two equal expansions for SG (f ) and therefore to a stable trace formula.
The proofs of the four identities are completed in the two papers [54] and [55]. They are similar
in many respects to those for base change. In particular, the most difficult case of the global geometrictheorem will be that for γ equal to the identity in the group. There is more to be said about it, but at thispoint my courage fails me. So I take farewell of the reader, wishing him, like an eminent predecessor
on another occasion, God-speed but leaving him to continue the journey on his own.

Bibliography of James G. Arthur[1] The Selberg trace formula for groups of F -rank one, Ann. of Math. 100 (1974), 326–385.
[2] Some tempered distributions on groups of real rank one, Ann. of Math. 100 (1974), 553–584.
[3] A theorem on the Schwartz space of a reductive Lie group, Proc. Nat. Acad. Sci. 72 (1975)
4718–4719.
[4] The characters of discrete series as orbital integrals, Inv. Math. 32 (1976), 205–261.
[5] A truncation process for reductive groups, Bull. Amer. Math. Soc. 83 (1977), 748–750.
[6] Eisenstein series and the trace formula, Proc. Sympos. Pure Math. 33, Part 1 (1979), 253–274.
[7] A trace formula for reductive groups I: Terms associated to classes in G(Q), Duke Math. J.
245 (1978), 911–952.
[8] Harmonic analysis of invariant distributions, Proc. Conf. on Lie Theories, (Queen’s University,1977), Queen’s Papers in Pure and Appl. Math 48 (1978), 384–393.
[9] A trace formula for reductive groups II: Applications of a truncation operator, Compositio
Math. 40 (1980) 87–121.
[10] The trace formula in invariant form, Ann. of Math. 114 (1981), 1–74.
[11] Automorphic representations and number theory, C.M.S. Conf. Proc. I (1980), 3–51.
[12] On the inner product of truncated Eisenstein series, Duke Math. J. 49 (1982), 35–70.
[13] On a family of distributions obtained from Eisenstein series I: Application of the PaleyWiener theorem, Amer. J. Math. 104 (1982), 1243–1288.

Introduction to the work of James Arthur

55[14] On a family of distributions obtained from Eisenstein series II: Explicit formulas, Amer. J.
Math. 104 (1982), 1289–1336.
[15] A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (l983), 1–89.
[16] The trace formula for reductive groups, ournées Automorphes, Publications Mathematiques de
L’Universite Paris VII 1983, 1–41.
[17] Multipliers and a Paley-Wiener theorem for real reductive groups, Representation Theory of
Reductive Groups, Birkhäuser 1983, 1–19.
[18] On some problems suggested by the trace formula, Lie Group Representations II, vol. 1041
Springer Lecture Notes in Math. (1984), 1–49.
[19] The trace formula for noncompact quotient, Proceedings of International Congress of Mathematicians (Warsaw, 1983) PWN, Warsaw, 1984, 849–859.
[20] A measure on the unipotent variety, Canad. J. Math. 37 (1985), 1237–1274.
[21] On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986), 179–214.
[22] The Fourier transform of weighted orbital integrals on SL(2, R) (with R. Herb and P. Sally),
Contemporary Math. 53 (1986), 17–37.
[23] The characters of supercuspidal representations as weighted orbital integrals, Proc. Indian
Acad. Sci. 97 (1987), 3–19.
[24] The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223–293.
[25] Characters, harmonic analysis, and an L2 -Lefschetz formula, Proc. Sympos. Pure Math. 48(1988), 167–179.
[26] The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), 323–383.
[27] The invariant trace formula. II. Global theory J. Amer. Math. Soc. 1 (1988), 501–554.
[28] Intertwining operators and residues. I. Weighted characters J. Funct. Anal. 84 (1989), 19–84.
[29] Intertwining operators and residues II. Invariant distributions, Compositio Math. 70 (1989),
51–99.
[30] Simple Algebras, Base Change and the Advanced Theory of the Trace Formula (with L. Clozel),
Annals of Math. Studies 120, Princeton University Press, 1989.
[31] The trace formula and Hecke operators, Number Theory, Trace Formulas and DiscreteGroups, Academic Press, 1989, 11–27.
[32] The L2 -Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257–290.
[33] Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one, in:Representation Theory and Harmonic Analysis on Semisimple Lie Groups (eds. P. Sally and D.
Vogan), Math. Surveys Monographs 31, Amer. Math. Soc., 1989, 13–100.
[34] Unipotent automorphic representations: Conjectures, Asterisque 171–172 (1989), 13–71.
[35] Unipotent automorphic representations: Global Motivation, in: Automorphic Forms, Shimura
Varieties and L-functions, vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10 (1989), 1–75.
[36] Towards a local trace formula, Proceedings of First JAMI Conference, Johns Hopkins University
Press, 1990, 1–23.

Introduction to the work of James Arthur

56[37] Lectures on automorphic L-functions (with S. Gelbart), L-functions and Arithmetic, (Durham,
1990), London Math. Soc. Lecture Note Ser. 153 (1991), 1–59.
[38] Some problems in local harmonic analysis, Harmonic Analysis on Reductive Groups (Brunswick,
ME, 1989), Progr. Math. 101 (1991), 57–78.
[39] A local trace formula, Pub. Math. I.H.E.S. 73, (1991), 5–96.
[40] On elliptic tempered characters, Acta Math. 171 (1993), 73–138.
[41] On the Fourier transforms of weighted orbital integrals, J. Reine Angew. Math. 452 (1994),
163–217.
[42] The trace Paley-Wiener theorem for Schwartz functions, Contemp. Math. 177 (1994), 171–180.
[43] L2 -cohomology and automorphic representations, in: Canadian Mathematical Society, 19451995, vol. 3, 1996, 1–17.
[44] On local character relations, Selecta Math. 2 (1996), 501–579.
[45] The problem of classifying automorphic representations of classical groups, CRM Proceedings
and Lecture Notes 11 (1997), 1–12.
[46] Stability and endoscopy: informal motivation, Proc. Sympos. Pure Math. 61 (1997), 433–442.
[47] Canonical normalization of weighted characters and a transfer conjecture, C. R. Math. Rep.
Acad. Sci., Canada 20 (1998), 35–52.
[48] Towards a stable trace formula, in: Proceedings of the International Congress of Mathematicians,
vol. II (Berlin, 1998), Doc. Math. 1998, Extra vol. II, 507–517.
[49] On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), 209–283.
[50] Endoscopic L-functions and a combinatorial identity, Canad. J. Math. 51 (1999), 1135–1148.
[51] Stabilization of a family of diﬀerential equations, Mathematical Legacy of Harish-Chandra(Baltimore, MD, 1998), Proc. Sympos. Pure Math. 68 (2000), 77–95.
[52] Harmonic analysis and group representations, Notices of AMS 47 (2000), 26–34.
[53] A stable trace formula I. General expansions, preprint.
[54] A stable trace formula II. Global descent, to appear in Inv. Math. .
[55] A stable trace formula III. Proof of the main theorems, preprint.
[56] The principle of functoriality, submitted to Mathematical Challenges of the 21st Century.

Supplementary references[ABV] J. Adams, D. Barbasch and D. Vogan, The Langlands Classiﬁcation and Irreducible Representations for Real reductive Groups, Progr. Math 104, Birkhäuser, 1992.
[CD] L. Clozel and P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lieréductifs, Inv. Math. 77 (1984), 427–453; Sur le théorème de Paley-Wiener invariant pourles groupes de Lie réductifs réels, C. R. Acad. Sci. Paris, Sér. 1 Math. 300 (1985), 331–333.
[HPS] R. Howe and I. Piatetski-Shapiro, A counterexample to the “generalized Ramanujan conjecture” for (quasi)-split groups, Automorphic forms, representations and L-functions (Corvallis, OR, 1977), Part 1, Proc. Sympos. Pure Math. 33 1979, 315–322.

Introduction to the work of James Arthur

57[JL] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes 114(1970).
[J] H. Jacquet, On the residual spectrum of GL(n), Lie group representations, II, Springer Lecture
Notes 1041, (1984), 185–208.
[K1] R. E. Kottwitz, Tamagawa numbers, Ann. of Math. 127 (1988), 629–646.
[K2], On the λ-adic representations associated to some simple Shimura varieties, Inv.
Math. 108 (1992), 653–665.

[KL] R. Langlands, Les débuts d’une formule des traces stables, Publ. Math. Univ. Paris VII
13, Paris, 1983; R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J.
, Stable trace formula: elliptic singular terms, Math. Ann. 275
51 (1984), 611–650;(1986), 365–399.
[KS] R. Kottwitz and D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 1999.
[L] R. P. Langlands, Rank-one residues of Eisenstein series, Israel Math. Conf. Proc. 3 (1990),
111–125.
[LL] J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979),
726–785.
[LS] R. P. Langlands and D. Shelstad, On the deﬁnition of transfer factors, Math. Ann. 278 (1987),
219–271.
[MW] C. Moeglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup.
22 (1989), 605–674.
[Mö] C. Moeglin, Représentations unipotentes et formes automorphes de carré intégrable, Forum
Math. 6 (1994), 651–744.
[Mü] W. Müller, The trace class conjecture in the theory of automorphic forms, Ann. of Math.
130 (1989), 473–529.
[R] J. Rogawski, Automorphic representations of unitary groups in three variables, Annals of
Math. Studies 123, Princeton University Press 1990; The multiplicity formula for A-packets,in The zeta functions of Picard modular surfaces, (eds. R. P. Langlands and D. Ramakrishnan),
Centre de recherches mathématiques, Univ. de Montréal, 1992, 395–419.
[S] H. Saito, Automorphic forms and extensions of number ﬁelds, Lectures in Mathematics 8,
Kyoto University, 1975.
[W] J.-L. Waldspurger, Le lemme fondamental implique le transfert, Comp. Math. 105 (1997),
153–236.