The Arthur-Selberg Trace formula

A superficial overview

Jean-Pierre Labesse

Université Paris 7

I – Presentation

The aim of these lectures is to give a brief account of the state of art for the Selberg trace formula as now established by Jim Arthur. The specialist may find this account extremely superficial and in fact it is! I found it impossible to give, in a few hours, a deeper description of the techniques involved for two reasons: firstly because of their intrinsic difficulty, and secondly because there is no reference available besides original articles for a work which goes over more than 25 years of intensive production. One cannot refer the reader to a more detailed but still penetrable account of the various techniques. The need for a rewriting of the whole story from the beginning is crying, but we shall not try to do that here.

Introduced in 1956 in his Bombay lecture, Selberg’s trace formula amounts to the equality of the geometric and of the spectral expansion, for the “renormalized trace” of the rightconvolution operator R(f) by a compactly supported smooth function f on G in L^{2}(Γ\G) where Γ is a discrete subgroup of finite co-volume in G. In general the trace of the convolution operator R(f) does not exist, this being due to the following fact: if the space Γ\G is of finite volume but not compact the spectral decomposition of L^{2}(Γ\G) is not discrete and has a continuous part in which the convolution operator cannot be of trace class unless it is zero. The strategy is to define a “renormalized trace” obtained via a cutoff of the divergent integral over the diagonal of the kernel associated to the convolution operator.

The co-compact case as well as the case of Γ = SL(2, ℤ) and G = SL(2, ℝ) had been treated by Selberg himself, but the case of a general arithmetic subgroup Γ in a real reductive group G (under the restriction that the pair (G, Γ) comes from an adelic situation) had to await for James Arthur’s work. In fact James Arthur has studied during the last 25 years the trace formula for connected reductive groups over the adeles i.e. the case where Γ = G(F) and G = G(𝔸_{F})^{1} where G is a connected reductive group defined over a number field F (see below for unexplained notation).

As can be expected the cutoff procedure destroys the invariance of the trace. The first form of the trace formula is an identity of non-invariant distributions. The invariance can eventually be restored and one such gets the invariant form of the trace formula. This allows to apply the trace formula to functions that are known only through their scalar Fourier transform, for example pseudo-coefficients of discrete series. This is what is used for the computation of traces of Hecke operators in the cohomology of locally symmetric spaces in particular for Shimura varieties.

Although the explicit computation of multiplicities of discrete series at achimedean places is of great historical importance, most of applications, like base change, are not through such explicit computations but rather through the comparison of two trace formulas on two different groups. Even in the study of Shimura varieties, the aim is not the explicit knowledge of the traces of Hecke operators in cohomology spaces but to make the comparison between the results given by Arthur-Selberg trace formula and the Lefschetz trace formula. This technique, due to Langlands, was first applied to the famous Jacquet-Langlands comparison between automorphic forms attached to quaternion algebras and automorphic forms on GL(2). For such comparisons one needs to put the trace formula into the so-called stable form. The trace formula is already in its stable form when one deals with GL(n) but for other groups (and even for GL(n) in the twisted case) this is a difficult problem.

James Arthur has solved the problems that were on the way to obtain a manageable expression for the trace formula in full generality. By manageable I understand an expression which is sufficiently explicit to allow a trace formula comparison between different groups, as expected by Langlands, to deal with some cases of functoriality (endoscopic and twisted endocopic functoriality) taking for granted the “Fundamental Lemma”. The most recent papers by Arthur deal with this question for arbitrary reductive groups. Unfortunately, the fundamental lemma remains unsettled in most cases.

There are some old survey articles [A2], [A3] or [L1]. The following notes are in part extracted from the survey [L2]. There are more recent surveys by Arthur e.g. his talks in Edinburgh (1996) or in Berlin (1998). A rather complete bibliography of Arthur’s work is given at the end of these notes.

II – The cocompact case

II.1 – The trace formula

Let G be a locally compact group and Γ a discrete subgroup. In this section we assume that Γ\G is compact; this imply that G is unimodular. We choose a Haar measure on G.

Consider f ∈ C_{c}(G) a continuous compactly supported function on G, and let R(f) be the operator defined by f acting via the right regular representation in L^{2}(Γ\G). The basic observation is that the operator R(f) is given by a smooth kernel K on Γ\G: for φ ∈ L^{2}(Γ\G) we have

(R(f)φ)(x) = ∫_{G} f(y) φ(xy) dy = ∫_{G} f(x^{−1}y) φ(y)dy = ∫_{Γ\G} K(x, y) φ(y) dy

with

K(x, y) = ∑_{γ∈Γ} f(x^{−1}γy) .

Since f si compactly supported, the kernel K is given by a continuous function. Since Γ\G is compact the kernel K is square integrable and hence R(f) is a Hibert-Schmidt operator. A theorem of Dixmier and Malliavin tells us that any smooth compactly supported function f is a finite sum of convolution products of compactly supported smooth functions g_{i} and h_{i}:

f = ∑_{i=1}^{i=r} g_{i} ∗ h_{i} ,

from this we conclude that the operator R(f) is a finite sum of products of Hilbert-Schmidt operators and hence is of trace class for f ∈ C_{c}^{∞}(G). One has two expressions for its trace.

First the trace can be computed by integrating the kernel K(x, y) over the diagonal:

trace R(f) = ∫_{Γ\G} K(x, x) dx = ∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γx) dx .

By expanding the kernel according to conjugacy classes in Γ we get:

∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γx) dx = ∑_{γ∈{Γ}} ∫_{Γ_{γ}\G} f(x^{−1}γx) dx .

where {Γ} is a set of representatives of conjugacy classes in Γ, and Γ_{γ} denotes the centralizer of γ in Γ. Let G_{γ} denote the centralizer of γ in G we introduce

a_{Γ}(γ) = vol (Γ_{γ}\G_{γ})

and

I_{G}(γ, f) = ∫_{G_{γ}\G} f(x^{−1}γx) dẋ

the orbital integral. This yields the geometric expansion of the trace formula in the cocompact case:

trace R(f) = ∑_{γ∈{Γ}} a_{Γ}(γ) I_{G}(γ, f) .

Observe that, thanks to the compactness of Γ\G all sums and integrals are absolutely convergent.

The second expression is given by the spectral decomposition of R. Since the operators R(f) are compact (in fact they are of Hibert-Schmidt type for all compactly supported smooth functions f), a standard argument shows that the representation (R, L^{2}(Γ\G)) splits into a discrete sum of unitary irreducible representations (π, V_{π}) with finite multiplicity a_{Γ}(π):

L^{2}(Γ\G) = ˆ⨁_{π∈Π(G)}(V_{π})^{a_{Γ}(π)}

where Π(G) is the unitary dual of G. Then

trace R(f) = ∑_{π∈Π(G)} a_{Γ}(π) I_{G}(π, f)

where the distribution

I_{G}(π, f) = trace π(f)

is the character of π .

The trace formula for the co-compact case is the identity:

∑_{γ∈{Γ}} a_{Γ}(γ) I_{G}(γ, f) = ∑_{π∈Π(G)} a_{Γ}(π) I_{G}(π, f)

for all f ∈ C_{c}^{∞}(G).

II.2 – First application

Consider the case where G is a connected semisimple real Lie group and let Γ be a discrete torsion free co-compact subgroup. At least when G is linear such Γ exist. Consider a function f whose semi-simple orbital integrals I_{G}(γ, f) vanish unless γ is elliptic (we say that γ is elliptic if its centralizer is a reductive subgroup and contains an elliptic cartan subgroup). Such a function is called ‘cuspidal’ by Arthur. For such a Γ and such a function f the geometric expansion of the trace formula reduces to one term:

trace R(f) = vol (Γ\G)f(1)

Let f_{e} be an “Index-function” associated to some auxilliary data: e = (P, ξ), as decribed in an appendix below. Index functions can be constructed using the scalar Paley-Wiener theorem [CD] or using Arthur-Delorme multipliers [D1]; we refer the reader to [L3] for a more detailed study of these functions.

We know that Index functions are cuspidal i.e. I_{G}(γ, f_{e}) = 0 if γ is semisimple unless γ is elliptic. For γ regular elliptic I_{G}(γ, f_{e}) = I(γ, e) can be computed explicitly but we shall not need that.

On the spectral side we know that I_{G}(π, f) = I(π, e) is zero unless π has some fixed infinitesimal character defined by e. The sum over all representations is then the index for a Dirac operator D_{e} acting on some fiber bundle E on Γ\G/K associated to e and hence the trace formula applied to f_{e} is the index formula:

vol (Γ\G)f_{e}(1) = Index(D_{e}, E) .

This formula is the starting point of the proof by Atiyah and Schmid of Harish Chandra’s theorem on the existence and classification of discrete series. In particular, if e defines a discrete series π_{e} with sufficiently regular parameter, the number Index(D_{e}, E) is noting but the multiplicity of the discrete series π_{e} in L^{2}(Γ\G).

6 J.-P. Labesse

III – The basic example: GL(2)

In this section G = GL(2) and P is the subgroup of upper triangular invertible 2 by 2 matrices. We have P = MN with M the subgroup of diagonal matrices and N the subgroup of unipotent upper triangular matrices. We work from now on in the adelic setting. The inexperienced reader should consult the appendix on adeles for an introduction.

III.1 – The geometric side for GL(2)

We associate to f ∈ C_{c}^{∞}(G(𝔸) two kernels

K(x, y) = ∑_{γ∈G(ℚ)} f(x^{−1}γy)

and

K_{P} (x, y) = ∑_{ξ∈M(ℚ)} ∫_{N(𝔸)} f(x^{−1}ξny)dn .

Let

𝔞_{P} = Lie(M) ⊗ ℝ .

The Iwasawa decomposition and the map from M(𝔸) to 𝔞_{P} induced by the logarithm of the absolute value of characters allows us to define a map

H_{P} : G(𝔸) → 𝔞_{P}

Let T be a point in 𝔞_{P}; we introduce (with the notation of the general case) the function τ̂_{P} (x, T) which is the characteristic function of the set of x ∈ G(𝔸) such that ϖ(H_{P} (x)−T) > 0 where ϖ is the fundamental weight. A truncated version of the restriction to the diagonal of K is the function

k^{T} (x) = K(x, x) − ∑_{δ∈P (ℚ)\G(ℚ)} τ̂_{P} (δx, T)K_{P} (δx, δx) .

Viewed as a function on the classical fundamental domain, i.e. if x = nak corresponds to z = u + it with

na = ( \begin{matrix}t & u \\ 0 & 1 \end{matrix} )

in the upper half plane with u = |Re(z)| < 1/2 and |z| > 1 and in the particular case where ϖ(T) > 0 we have

k^{T} (x) = K(x, x) if t ≤ e^{ϖ(T)} and

k^{T} (x) = K(x, x) − K_{P} (x, x) if t > e^{ϖ(T)}.

The first result is the

III.1. Theorem. – The function k^{T} is integrable on GL(2, ℚ)\GL(2, 𝔸)^{1}.

Hence one can define

J^{T} (f) = ∫_{G(ℚ)\G(𝔸)^{1}} k^{T} (x)dx .

The proof of the convergence is obtained by an almost direct computation based on the splitting of the kernel K according to conjugacy classes. One has

K = K_{e} + K_{s} + K_{u}

where K_{e} is the kernel obtained by summation in Γ = G(ℚ) over the set Γ_{e} of elliptic elements, K_{s} by summation over semisimple non-elliptic elements (i.e. split non central), and K_{u} by sommation over unipotent non central elements. One obtains by restriction to the diagonal a decomposition

k^{T} (x) = k_{e}(x) + k_{s}^{T} (x) + k_{u}^{T} (x)

and by integration we get

J^{T} (f) = J_{e}(f) + J_{s}^{T} (f) + J_{u}^{T} (f)

The elliptic part of the kernel is integrable without any truncation: the integral

J_{e}^{T} (f) = ∫_{G(ℚ)\G(𝔸)^{1}} k_{e}(x)dx

is absolutely convergent and can be rewritten as in the co-compact case

J_{e}^{T} (f) = ∑_{{Γ_{e}}} vol (G_{γ}(ℚ)\G_{γ}(𝔸)^{1}) ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γx)dx

using that

G_{γ}(𝔸)^{1}\G(𝔸)^{1} ≃ G_{γ}(𝔸)\G(𝔸)

where {Γ_{e}} is a set of representatives of conjugacy classes of elliptic elements.

The Arthur-Selberg Trace Formula 9

The divergent contribution comes from split and unipotent elements; they become convergent after truncation. The truncated contribution of split elements can be written

J_{s}^{T} (f) = ∫_{G(ℚ)\G(𝔸)^{1}}[K_{s}(x, x) − ∑_{δ∈P (ℚ)\G(ℚ)} τ̂_{P} (δx, T)K_{P}^{∗} (δx, δx)]dx

where

K_{P}^{∗} (x, x) = ∑_{ξ∈M(ℚ)^{∗}} ∫_{N(𝔸)} f(x^{−1}ξnx)dn = ∑_{ξ∈M(ℚ)^{∗}} ∫_{N(𝔸)} f(x^{−1}n^{−1}ξnx)dn

where M(ℚ)^{∗} is the set of non central diagonal rational elements. Since

K_{s}(x, x) = ∑_{δ∈M(ℚ)\G(ℚ)} \frac{1}{2} ∑_{ξ∈M(ℚ)^{∗}} f(x^{−1}δ^{−1}ξδx)

the integral defining J_{s}^{T} (f) can be written

J_{s}^{T} (f) = ∫_{M(ℚ)\G(𝔸)^{1}} \frac{1}{2} ∑_{ξ∈M(ℚ)^{∗}}[1 − τ̂_{P} (x, T) − τ̂_{P} (wx, T)]f(x^{−1}ξx)dx .

This is again equal to

J_{s}^{T} (f) = \frac{1}{2} ∑_{ξ∈M(ℚ)^{∗}} vol (M(ℚ)\M(𝔸)^{1}) ∫_{M(𝔸)\G(𝔸)} f(x^{−1}ξx)v_{M}^{T}(x)dx

with

v_{M}^{T}(x) = ∫_{A_{G}M(𝔸)^{1}\M(𝔸)}[1 − τ̂_{P} (mx, T) − τ̂_{P} (wmx, T)] dm .

We observe that if we put v_{M} = v_{M}^{0} we have

v_{M}^{T} = v_{M} + c(T)

where c is a linear form. The remaining term is

J_{u}^{T} (f) = ∑_{ζ∈Z} ∫_{P (ℚ)\G(𝔸)^{1}} [ ∑_{η∈N(ℚ)^{∗}} f(x^{−1}ζηx) − τ̂_{P} (x, T) ∫_{N(𝔸)} f(x^{−1}ζnx)dn] dx

whose convergence follows from the Poisson summation formula. A more tractable expression is obtained via an other regularisation. In any case this is a polynomial of degree 1 in T.

The geometric side of the trace formula is a polynomial of the first degree in T:

J^{T} (f) = J(f) + c_{1}(T)

where c_{1}(T) is a linear form. Let

a^{G}(γ) = vol (G_{γ}(ℚ)\G_{γ}(𝔸)^{1})  and  a^{M}(ξ) = vol (M_{ξ}(ℚ)\M_{ξ}(𝔸)^{1})

and

J_{G}(γ, f) = ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γx)dx  and  J_{M}(ξ, f) = ∫_{M_{ξ}(𝔸)\G(𝔸)} f(x^{−1}ξx)v_{M}(x) dx

with such notation we get

J(f) = ∑_{γ∈\{G_{e}(ℚ)\}} a^{G}(γ) J_{G}(γ, f) + \frac{1}{2} ∑_{ξ∈M(ℚ)^{∗}} a^{M}(ξ)J_{M}(ξ, f) + J_{u}(f) .

III.2 – Truncated Eisenstein series for GL(2)

In this subsection we compute the scalar product of truncated Eisenstein series for G = GL(2).

Let P be the subgroup of invertible upper triangular matrices. Let ϕ be a smooth function on G(𝔸) such that

ϕ(px) = ϕ(x) for p ∈ A_{P} M(ℚ)N(𝔸) ⊂ P (𝔸)

and compactly supported modulo A_{P} M(ℚ)N(𝔸). To such a function we associate for each λ ∈ 𝔞_{P}^{∗} a new function

ϕ(x, λ) = exp (< λ + ρ_{P} , H_{P} (x) >) ϕ(x)

where ρ_{P} is the half sum of the positive roots. This allows to define an Eisenstein series

E(x, ϕ, λ) = ∑_{γ∈P (ℚ)\G(ℚ)} ϕ(γ x, λ)

The Arthur-Selberg Trace Formula 11

by mean of this series convergent for Reλ > ρ_{P}. Its “constant term” along P is given by

E^{0}(x, ϕ, λ) = ϕ(x, λ) + ∫_{N(𝔸)} ϕ(wnx, λ) dn = ϕ(x, λ) + (M(λ)ϕ)(x, λ^{w})

where (M(λ)ϕ)(x) = ∫_{N(𝔸)} exp (< λ + ρ_{P} , H_{P} (wnx) > − < λ^{w} + ρ_{P} , H_{P} (x) >) ϕ(wnx) dn

The truncated Eisenstein series Λ^{T} E(x, ϕ, λ) (denoted E′′(x, ϕ, λ) in Selberg’s and Langlands’ articles on Eisenstein series) is defined by

Λ^{T} E(x, ϕ, λ) = E(x, ϕ, λ) − ∑_{δ∈P (ℚ)\G(ℚ)} τ̂_{P} (δx, T)E^{0}(δx, ϕ, λ)

We need an expression for the scalar product of two truncated Eisenstein series. ω^{T} (ϕ, ψ, λ, μ) = ∫_{G(ℚ)\G(𝔸)^{1}} Λ^{T} E(x, ϕ, λ) \overline{Λ^{T} E(x, ψ, μ)} dx = ∫_{G(ℚ)\G(𝔸)^{1}} Λ^{T} E(x, ϕ, λ) \overline{E(x, ψ, μ)} dx This can be rewritten

ω^{T} (ϕ, ψ, λ, μ) = ∫_{P (ℚ)\G(𝔸)^{1}} Λ^{T} E(x, ϕ, λ) \overline{ψ(x, μ)} dx

provided Reμ > ρ_{P}, and now we get ω^{T} (ϕ, ψ, λ, μ) = ∫_{P (ℚ)N(𝔸)\G(𝔸)^{1}} (Λ^{T} E(x, ϕ, λ))^{0} \overline{ψ(x, μ)} dx

where (Λ^{T} E(x, ϕ, λ))^{0} is the constant term along P. But for Re(Λ) > ρ_{P}

(Λ^{T} E(x, ϕ, λ))^{0} = [1 − τ̂_{P} (x, T)]E^{0}(x, ϕ, λ) + ∫_{N(𝔸)} τ̂_{P} (wnx, T) E^{0}(wnx, ϕ, λ) dn and hence, for Re(μ) > |Re(Λ)|, A(ϕ, ψ, λ, μ) is the sum of two terms:

∫_{P (ℚ)N(𝔸)\G(𝔸)^{1}}[1 − τ̂_{P} (x, T)]E^{0}(x, ϕ, λ) \overline{ψ(x, μ)} dx

= ∫_{𝔞_{P}^{G}}[1 − τ̂_{P} (X − T)] [< ϕ, ψ > e^{<λ+\overline{μ},X>}+ < M(λ)ϕ, ψ > e^{<λ^{w}+\overline{μ},X>}] dX

12 J.-P. Labesse

and

∫_{P (ℚ)N(𝔸)\G(𝔸)^{1}} ∫_{N(𝔸)} τ̂_{P} (wnx, T) E^{0}(wnx, ϕ, λ) \overline{ψ(x, μ)} dn dx
= ∫_{M(ℚ)\G(𝔸)^{1}} τ̂_{P} (wx, T) E^{0}(wx, ϕ, λ) \overline{ψ(x, μ)} dx
= ∫_{M(ℚ)\G(𝔸)^{1}} τ̂_{P} (x, T) E^{0}(x, ϕ, λ) \overline{ψ(wx, μ)} dx

= ∫_{𝔞_{P}^{G}} τ̂_{P} (X − T) [< ϕ, M(μ)ψ > e^{<λ+\overline{μ^{w}},X>}+ < M(λ)ϕ, M(μ)ψ > e^{<λ^{w}+\overline{μ^{w}},X>}] dX and we get

ω^{T} (ϕ, ψ, λ, μ) = \frac{< ϕ, ψ > e^{<λ+\overline{μ},T >}− < M(λ)ϕ, M(μ)ψ > e^{<λ+\overline{μ},w(T)>}}{< λ + \overline{μ}, T_{0} >}

+ \frac{< M(λ)ϕ, ψ > e^{<λ^{w}+\overline{μ},T >}− < ϕ, M(μ)ψ > e^{<λ^{w}+\overline{μ},w(T)>}}{< λ^{w} + \overline{μ}, T_{0} >}

for some T_{0} depending on the choice of Haar measures. Then for λ = μ we have

ω^{T} (ϕ, ψ, λ, λ) = \frac{< ϕ, ψ > e^{2<Re(λ),T >}− < M(λ)ϕ, M(λ)ψ > e^{2<Re(λ),w(T)>}}{2 < Re(λ), T_{0} >}

− \frac{< M(λ)ϕ, ψ > e^{−2i< Im (λ),T >}− < ϕ, M(λ)ψ > e^{−2i< Im (λ),w(T)>}}{2i < Im (λ), T_{0} >} .

This expression has limit on the imaginary axis Re(λ) = 0. Using that on the imaginary axis

M(λ)^{∗} = M(λ)^{−1} = M(−λ)

we get for T = tT_{0} ω^{T} (ϕ, ψ, λ, λ) = 2t < ϕ, ψ > + < M(λ)^{−1}M′(λ)ϕ, ψ >

+ \frac{< M(−λ)ϕ, ψ > e^{2it< Im (λ),T_{0}>}− < M(λ)ϕ, ψ > e^{−2it< Im (λ),T_{0}>}}{2i < Im (λ), T_{0} >}

III.3 – The spectral side for GL(2) Let K_{cusp} be the restriction of the kernel to the cuspidal spectrum, K_{res} the restriction of the kernel to the residual spectrum and R_{λ} the right regular representation in the normalized induced representation from P (𝔸) to G(𝔸) of the regular representation of M(𝔸)^{1} in L^{2}(M(ℚ)\M(𝔸)^{1}) extended to P (𝔸) trivially on A_{G}N(𝔸) and twisted by λ on 𝔞_{P} /𝔞_{G} ≃ A_{P} /A_{G}. Let B_{λ} be an orthonormal basis of L^{2}(M(ℚ)\M(𝔸)^{1}). We have

K(x, y) = K_{cusp}(x, y) + K_{res}(x, y) + \frac{1}{2} ∫_{i(𝔞_{P}^{G})^{∗}} ∑_{ψ∈B_{λ}} E(x, R_{λ}(f)ψ, λ)\overline{E(y, ψ, λ)} dλ

where dλ is the measure dual to the measure on 𝔞_{P}^{G}. Hence

Λ^{T} KΛ^{T} (x, y) = K_{cusp}(x, y)+Λ^{T} K_{res}Λ^{T} (x, y)+\frac{1}{2} ∫_{i(𝔞_{P}^{G})^{∗}} ∑_{ψ∈B_{λ}} Λ^{T} E(x, R_{λ}(f)ψ, λ)\overline{Λ^{T} E(y, ψ, λ)} dλ

so that the trace of this operator is given by

trace Λ^{T} KΛ^{T} = trace K_{cusp} + trace Λ^{T} K_{res}Λ^{T}

+\frac{1}{2} ∫_{G(ℚ)\G(𝔸)^{1}} ∫_{i(𝔞_{P}^{G})^{∗}} ∑_{ψ∈B_{λ}} Λ^{T} E(x, R_{λ}(f)ψ, λ)\overline{Λ^{T} E(x, ψ, λ)} dλ dx

and hence

trace Λ^{T} KΛ^{T} = trace K_{cusp} + trace Λ^{T} K_{res}Λ^{T} + \frac{1}{2} ∫_{i(𝔞_{P}^{G})^{∗}} ∑_{ψ∈B_{λ}} ω^{T} (R_{λ}(f)ψ, ψ, λ, λ) dλ

We observe that

trace Λ^{T} K_{res}Λ^{T} = trace K_{res} + o_{1}(T)

and we get

trace Λ^{T} KΛ^{T} = J_{disc}(f) + \frac{1}{2} ∫_{i(𝔞_{P}^{G})^{∗}} trace (M(λ)^{−1}M′(λ)R_{λ}(f)) dλ + c_{2}(T) + o_{2}(T)

where c_{2}(T) is a linear form and

J_{disc}(f) = trace R_{disc}(f) + \frac{1}{4} trace (M(0)R_{0}(f))

is called the “discrete part” of the spectral side of the trace formula. We observe that besides the contribution of the discrete spectrum, there is a contribution of the Eisenstein series.

Now, for T large enough (depending on f), we have

k^{T} (x) = Λ^{T} KΛ^{T} (x, x)  and hence  J^{T} = trace Λ^{T} KΛ^{T} .

This shows that

J(f) = J_{disc}(f) + \frac{1}{2} ∫_{i(𝔞_{P}^{G})^{∗}} trace (M(λ)^{−1}M′(λ)R_{λ}(f)) dλ .

14 J.-P. Labesse

IV – General trace formula: the geometric side

IV.1 – The truncated kernel over the diagonal

We return now to the general case. We fix a minimal parabolic subgroup P_{0} with Levi subgroup M_{0} ; in the sequel we shall consider only parabolic subgroups P with Levi subgroup M_{P} such that M_{0} ⊂ M_{P} .

Let P be a parabolic subgroup of G and let X(P) be the group of its rational characters; we denote by 𝔞_{P} the vector space Hom_{ℤ}(X(P), ℝ) . There is a natural map:

H_{P} : P (𝔸_{F}) ⟶ 𝔞_{P}

whose kernel is P (𝔸_{F})^{1}. If P ⊂ Q we denote by 𝔞_{P}^{Q} the kernel of the natural map 𝔞_{P} → 𝔞_{Q} . The map H_{P} has a section whose image lies in the center of a Levi subgroup M_{P} (F ⊗ ℝ) of P (F ⊗ ℝ), and will be denoted A_{P} . This allows to define sections for the maps 𝔞_{P} → 𝔞_{Q} .

Then one defines τ̂_{P} as the caracteristic function of the the open cone in 𝔞_{P}^{G} generated by Δ̌_{P} the simple co-roots of the pair (P, S_{P}) where S_{P} is the split component of the center of M_{P} . We shall also need the functions τ̂_{P}^{Q} on 𝔞_{P}^{Q} defined using co-roots of (P ∩ M_{Q}, S_{P}), and the functions of two variables Γ_{Q}(H, X) on 𝔞_{Q}^{G} × 𝔞_{Q}^{G} such that

τ̂_{P} (H − X) = ∑_{Q⊃P}(−1)^{a_{Q}^{G}} τ̂_{P}^{Q}(H) Γ_{Q}(H, X)

where a_{Q}^{G} is the dimension of 𝔞_{Q}^{G} .

Choose a ”good” maximal compact subgroup K ⊂ G(𝔸_{F}) then by the Iwasawa decomposition x ∈ G(𝔸_{F}) can be written x = a_{P} (x) p k with p ∈ P (𝔸_{F})^{1} , k ∈ K and where a_{P} (x) ∈ A_{P} is uniquely defined. Then H_{P} can be extended to a map on G(𝔸_{F}) such that H_{P} (x) = H_{P} (a_{P} (x)) .

Given T ∈ 𝔞_{0} sufficiently deep in the positive Weyl chamber one can define [A1a] the cutoff trace denoted J^{T} (f) – or J^{T,G}(f) if one wants to emphasize the dependance on G :

J^{T} (f) = ∫_{G(ℚ)\G(𝔸_{F})^{1}} k^{T} (x) dx

where

k^{T} (x) = ∑_{P ⊃P_{0}}(−1)^{a_{P}^{G}} ∑_{δ∈P (ℚ)\G(ℚ)} τ̂_{P} (H_{P} (δx) − T) K_{P} (δx, δx)

with a_{P}^{G} = dim 𝔞_{P}^{G} and

K_{P} (x, y) = ∑_{γ∈M_{P} (ℚ)} ∫_{N_{P} (𝔸_{F})} f(x^{−1}γny) dn .

As a function of T the cutoff trace defined on an open set of 𝔞_{0} is a polynomial. In fact if one changes the cutoff parameter T to T + X then one sees easily that

J^{T +X}(f) = ∑_{Q⊃P_{0}} J^{T,Q}(f_{Q})γ_{Q}(X)

where J^{T,Q}(f_{Q}) = J^{T,M_{Q}}(f_{Q}) is the cutoff trace for M_{Q} applied to

f_{Q}(m) = δ_{Q}(m)^{1/2} ∫_{K} ∫_{N_{Q}(𝔸_{F})} f(k^{−1}mnk) dk dn

here δ_{Q} is the module of Q(𝔸_{F}) and

γ_{Q}(X) = ∫_{𝔞_{Q}^{G}} Γ_{Q}(H, X) dH

is a polynomial in X ; when X is ”positive” this is the volume of a compact convex subset in 𝔞_{Q}/𝔞_{G} .

The cutoff trace J^{T} can then be continued to all T ∈ 𝔞_{0} as a polynomial. One defines

T_{0} = \frac{1}{w^{G}} ∑_{s∈W^{G}} H_{P_{0}}(w_{s}^{−1})

where w_{s} is a representative in G(ℚ) of s in W^{G} the Weyl group of G , and w^{G} the cardinal of W^{G} . The element T_{0} is such that

T_{0} − s^{−1}T_{0} = H_{P_{0}}(w_{s}^{−1}) .

The value at T = T_{0} of J^{T,Q} will be denoted J^{Q} or even simply J when Q = G . Even at T = T_{0} the cutoff trace is not invariant, the distribution J still depends on the choice of M_{0} and K . More precisely let f^{y}(x) = f(yxy^{−1}) and for the sake of simplicity assume that f is K-central that is f^{k} = f for all k ∈ K then a simple computation shows that:

J(f^{y}) = ∑_{Q⊃P_{0}} J^{Q}(f_{Q})u_{Q}(y)

where

u_{Q}(y) = ∫_{K} ∫_{𝔞_{Q}^{G}} Γ_{Q}(X, −H_{Q}(ky)) dk dX .

IV.2 – The geometric expansion

The geometric term is obtained by expanding J(f) as a sum over 𝔒 the set of equivalence classes of elements in G(F) the equivalence being the conjugacy of their semisimple part:

J(f) = ∑_{𝔬∈𝔒} J_{𝔬}(f)

where J_{𝔬}(f) is defined as J(f) above but with K_{P} replaced by K_{P,𝔬} itself defined by

K_{P,𝔬}(x, y) = ∑_{γ∈M_{P} (F)∩𝔬} ∫_{N_{P} (𝔸_{F})} f(x^{−1}γny) dn .

If 𝔬 is the class of a semisimple element γ whose centralizer is connected and anisotropic modulo its center, then J_{𝔬} has a simple expression given by a weighted orbital integral: let M be a Levi subgroup that intersects 𝔬 and is minimal for this property; we may assume that M contains γ , then G_{γ} = M_{γ} and

J_{𝔬}(f) = a^{M}(γ) J_{M}(γ, f)

where

a^{M}(γ) = vol (M_{γ}(F)\M_{γ}^{1}(𝔸_{F}))

and

J_{M}(γ, f) = ∫_{G_{γ}(𝔸_{F})\G(𝔸_{F})} f(x^{−1}γx) v_{M}(x) dẋ

is a weighted orbital integral. The weight v_{M} is defined as follows:

v_{M}(x) = ∑_{Q∈P(M)} γ_{Q}(T_{0} − H_{Q}(x))

where the sum is over P(M) the set of parabolic subgroups with Levi subgroup M .

But in general there does not seem to be any simple explicit expression for J_{𝔬}(f) when the γ ∈ 𝔬 may have non trivial unipotent parts: already for SL(2) one must first stabilize the trace formula to get reasonably simple expressions for the contribution of unipotent elements, like those for GL(2) ; moreover for groups of higher rank the geometry of nilpotent orbits comes into play to mess up the situation. Nevertheless for a restricted set of functions f the distribution J_{𝔬}(f) can be shown to be a linear combination of generalized weighted orbital integrals that will be defined below.

Let S be a finite set of places of F and let f be a function in C_{c}^{∞}(G(F_{S})^{1}) where F_{S} is the product of completions of F over places in S . One can define [A6] a generalized notion of weighted orbital integral for all γ ∈ M(F_{S}) . Assume first that G_{γ} = M_{γ} , and let G_{γ}^{0} be the neutral component of G_{γ} , then introduce

J_{M}(S, γ, f) = |D(S, γ)|^{1/2} ∫_{G_{γ}^{0}(F_{S})\G(F_{S})} f(x^{−1}γx) v_{M}(S, x) dẋ

with

D(S, γ) = ∏_{v∈S} det {(1 − adγ_{s})|𝔤/𝔤_{γ_{s}}}_{v}

here 𝔤 is the lie algebra of G and γ_{s} is the semi simple part of the Jordan decomposition γ = γ_{s}γ_{u}; the weight is

v_{M}(S, x) = ∑_{Q∈P(M)} γ_{Q}(T_{0,S} − H_{Q}(x_{S})) .

In general one defines J_{M}(γ, f) as a limit over regular a ∈ A_{M} of a sum over L(M) the set of Levi subgroups containing M :

J_{M}(S, γ, f) = lim_{a→1} ∑_{L∈L(M)} r_{M}^{L}(S, γ, a) J_{L}(S, aγ, f)

where the r_{M}^{L}(S, γ, a) are themselves defined as limits over regular λ ∈ 𝔞_{M}^{∗} of sums over P^{L}(M) the set of parabolic subgroups with Levi subgroup M :

r_{M}^{L}(S, γ, a) = lim_{λ→0} ∑_{R∈P^{L}(M)} r_{R}^{L}(S, λ, γ, a) θ_{R}(λ)^{−1}

with

θ_{R}(λ) = vol (𝔞_{R}/ℤ(Δ̌_{R}))^{−1} ∏_{α∈Δ_{R}} λ(α̌)

and

r_{R}^{L}(S, λ, γ, a) = ∏_{v∈S} ∏_{β∈Δ_{R}^{L}} |a^{β} − a^{−β}|_{v}^{λ(β̌)R(β,γ_{u})}

the R(β, γ_{u}) being so chosen that the limits exist and do not vanish identically.

Now assume that f is the product of a function in C_{c}^{∞}(G(F_{S_{0}})^{1}) by the characteristic function of the maximal compact subgroup outside S_{0} a finite set of places containing archimedean ones; it can be shown [A7] that the distributions J_{𝔬} are linear combinations of S-weighted orbital integrals provided that S ⊃ S_{0} is big enough.

This yields the so called ”fine 𝔒 expansion”:

IV.1. Proposition. – Assume that S ⊃ S_{0} is big enough, then there exist uniquely defined numbers a^{L}(S, γ) such that

J(f) = ∑_{L∈L} \frac{w^{L}}{w^{G}} ∑_{γ∈\{L(F)\}_{L,S}} a^{L}(S, γ) J_{L}(S, γ, f)

where {L(F)}_{L,S} is a set of representatives for γ ∈ L(F) up to conjugacy under L(F) and for fixed semi simple part γ_{s} up to conjugacy under L_{γ_{s}}(F_{S}) of the unipotent part of γ .

The existence proof for the coefficients a^{L}(S, γ) proceeds by induction on the Levi subgroups L ∈ L . But it does not give in general a procedure to compute them explicitly. Happily the following particular case seems to be enough for many applications. We need first a definition: consider γ semisimple and contained in a Levi subgroup M, we shall say that γ is M-elliptic if its M-centralizer M_{γ} contains a torus maximal in M and anisotropic modulo the center. Let i^{M}(γ) be the number of connected components of M_{γ} with rational points.

IV.2. Proposition. – Assume γ is semisimple and S big enough then

a^{L}(S, γ) = i^{L}(γ)^{−1} vol (L_{γ}^{0}(F)\L_{γ}^{0}(𝔸_{F})^{1})

if γ is L-elliptic and equals 0 otherwise.

20 J.-P. Labesse

V – General trace formula: the spectral side

V.1 – The truncation operator

The truncation operator is defined formally as follows: let φ be a function on G(F)\G(𝔸_{F}) and let T ∈ 𝔞_{0} sufficiently regular with respect to the Weyl chamber defined by the choice of P_{0}. We put

Λ^{T} φ(x) = ∑_{P ⊃P_{0}}(−1)^{a_{P}^{G}} ∑_{δ∈P (F)\G(F)} τ̂_{P} (H_{P} (δx) − T) φ_{P} (δx)

and

φ_{P} (x) = ∫_{N_{P} (F)\N_{P} (𝔸_{F})} φ(nx) dn .

Since the summations are over finite sets uniformly for x in a compact set, and integrations are over compact sets, this make sense under mild regularity conditions on φ. The following lemma is the key of all properties:

V.1. Lemma. –

τ̂_{Q}(H_{Q}(x) − T)(Λ^{T} φ)_{Q}(x) = 0

unless Q = G.

Proof: We shall prove the lemma only when Q = P_{0} is a minimal parabolic subgroup. Recall that T is sufficiently regular. Assume τ̂_{Q}(H_{Q}(x) − T) ≠ 0 then τ̂_{P} (H_{P} (δx) − T) = 0 unless δ ∈ P (F). The alternating sum

∑_{P ⊃P_{0}}(−1)^{a_{P}^{G}}

vanishes thanks to the binomial identity.

□

The main properties are summed up in the next theorem.

V.2. Theorem. – The operator φ ↦ Λ^{T} φ is such that:

1 – Cuspidal automorphic forms are preserved.

2 – It extends to a self adjoint projector in L^{2}(G(F)\G(𝔸_{F})).

3 – Smooth functions with uniform moderate growth are transformed into rapidly decreasing functions.

Proof: The first assertion is clear. The second is seen as follows: the truncation operator with domain of definition the space of piecewise continuous rapidly decreasing function is symmetric. The above lemma shows that (Λ^{T})^{2} = Λ^{T} and hence Λ^{T} extends to an orthogonal projection operator. For the last assertion we give the proof in the case G = GL(2). It suffices to estimate |Λ^{T} (φ)(a)| for |a^{α}| large. But

Λ^{T} (φ)(a) = φ(a) − φ^{0}(a) = ∑_{ξ≠0} φ(a, ξ)

where φ(a, ξ) is the Fourier coefficient of index ξ:

φ(a, ξ) = ∫_{N(ℚ)\N(𝔸)} φ(ua) ψ(ξ(u))

here ξ is an additive character of 𝔸/ℚ. Now

|φ(x, ξ)| ≤ |ξ|^{−k}|a(x)^{α}|^{−k} sup_{u} |(φ ∗ X)(ux)|

where X is a generator of the Lie algebra of N. By hypothesis there is a constant C(X) such that

|(φ ∗ X)(x)| ≤ C(X) |a(x)^{α}|^{m}

for some integer m independent of X and hence

|Λ^{T} (φ)(a)| ≤ ∑_{ξ≠0} |φ(a, ξ)| ≤ C(X) |a^{α}|^{m−k} ∑_{ξ≠0} |ξ|^{−k} = C(X, k) |a^{α}|^{m−k}

for |a^{α}| large.

□

V.3. Corollary. – For f ∈ C_{c}^{∞}(G(𝔸_{F}) for φ ∈ L^{2}(G(F)\G(𝔸_{F})), the function Λ^{T} R(f)φ is rapidly decreasing.

Proof: Given φ ∈ L^{2}(G(F)\G(𝔸_{F})), the function R(f)φ = φ ∗ f̌ is of uniform moderate growth; in fact one has

|(φ ∗ f ∗ X)(x)| ≤ C_{1}(X, f)||φ||_{2} ||x||^{m}

for some m. The assertion follows from the previous theorem.

□

V.4. Theorem. – For f ∈ C_{c}^{∞}(G(𝔸_{F}), the operator Λ^{T} R(f) is of Hilbert-Schmidt type.

Proof: For the proof we restrict ourselves to the GL(2) case; following the proof of assertion 3 in the previous theorem, we observe that the estimate for |(φ ∗ f ∗ X)(x)| implies that there is a square integrable function Ξ such that

|Λ^{T} (φ ∗ f)(x)| ≤ Ξ(x) ||φ||_{2} .

The Schwarz inequality shows that the kernel K^{T} (x, y) of the operator Λ^{T} R(f) verifies

(∫ |K^{T} (x, y)|^{2} dy)^{1/2} ≤ Ξ(x)

and hence

∫ ∫ |K^{T} (x, y)|^{2} dx dy ≤ ∫ Ξ(x)^{2}dx < +∞ .
□

V.5. Corollary. – The operator Λ^{T} R(f)Λ^{T} is of trace class.

Proof: By the Dixmier-Malliavin’s theorem the function f is a finite sum of convolution products

f = ∑_{i} g_{i} ∗ h_{i}

and hence

Λ^{T} R(f)Λ^{T} = ∑_{i} Λ^{T} R(g_{i})R(h_{i})Λ^{T}

is a finite sum of trace class operators.

V.6. Theorem. – For T large enough the trace of the operator Λ^{T} R(f)Λ^{T} equals J^{T} (f):

J^{T} (f) = trace Λ^{T} R(f)Λ^{T} = ∫ K^{T} (x, x) dx .

V.2 – Intertwining operators

Before studying the spectral term we need to review some facts on intertwining operators. Given two parabolic subgroups P and Q and s ∈ W such that s(𝔞_{P}) = 𝔞_{Q} one defines for ϕ on N_{P} (𝔸_{F})M_{P} (F)\G(𝔸_{F})

(M_{Q|P} (s, λ)ϕ)(x) = ∫_{N(Q|P,s;𝔸_{F})} ϕ(w_{s}^{−1}nx) β(w_{s}^{−1}nx, x) dn

where N(Q|P, s) = N_{Q} ∩ w_{s}N_{P} w_{s}^{−1}\N_{Q} and

β(y, x) = exp(< λ + ρ_{P} , H_{P} (y) > − < sλ + ρ_{Q}, H_{Q}(x) >)

with ρ_{P} the half sum of roots of P in N_{P} . This converges for good ϕ and Re(λ) in a certain chamber and can be meromorphically continued.

We shall now define multidimensional logarithmic derivatives of intertwining operators [A4]. Consider

M_{Q}(P, λ, Λ) = M_{Q|P} (1, λ)^{−1}M_{Q|P} (1, λ + Λ)e^{Λ(T_{0})}

now assume Q ⊂ R and restrict λ and Λ to be in 𝔞_{R} then M_{Q}(P, λ, Λ) is independent of Q and will be denoted M_{R}(P, λ, Λ) . Let L be a Levi subgroup, then the logarithmic derivative is defined to be

M_{L}(P, λ) = lim_{Λ→0} ∑_{R∈P(L)} M_{R}(P, λ, Λ)θ_{R}(Λ)^{−1} .

Given π ∈ Π(M_{P}) and λ ∈ 𝔞_{M_{P}}^{∗} we denote by π_{λ} the representation of P (𝔸_{F}) defined by π_{λ}(nam) = a^{λ}π(m) and by I_{P} (π_{λ}) the unitarily induced representation from P (𝔸_{F}) up to G(𝔸_{F}) by π_{λ} . The restriction of operators M_{Q|P} (1, λ) to the subspace of vectors in I_{P} (π_{λ}) will be denoted M_{Q|P} (π_{λ})

Intertwining operators M_{Q|P} (π_{λ}) are the meromorphic continuation of the product over all places v of F of local operators. At a place v the local operator can be written as a product of a scalar function, the normalizing factor n_{Q|P} (π_{λ}) meromorphic in λ, and a ”normalized” intertwining operator R_{Q|P} (π_{λ})_{v} .

The normalized operators satisfy stronger functional equations than the ordinary ones (the cocycle relation will hold with no condition on the length for Weyl group elements). Moreover the following properties should hold:

– R_{Q|P} (π_{λ})_{v} is a rational function of λ if the place is archimedean or of {q_{v}^{<λ,α̌>} | α ∈ Δ_{P} } if the place is finite,

– the normalizing factor n_{Q|P} (π_{λ}) has neither zero nor pole if π is tempered and Re(λ) is in the positive Weyl chamber defined by P ;

– at finite places v where G and π are unramified and K_{v} hyperspecial, the normalized operator induces the identity on K_{v}-fixed vectors.

For archimedean places, using results of [KS], or when G = GL(n) thanks to [S1] the normalizing factors can be chosen as quotients of L-functions. In general the existence of normalized operators has been proved by Langlands (see [CLL] Lecture 15 and [A8]). But, for some purpose, the factors constructed by Langlands are not sufficiently ‘canonical’ and in the most recent papers, Arthur uses an other normalization based on the Plancherel measure; although this is not strictly speaking a normalization (the cocycle relation may not hold) it is canonical and more suited to comparison of trace formulas on different groups.

Using normalized operators instead of ordinary ones logarithmic derivatives R_{L}(P, π_{λ}) are defined by the same procedure as M_{L}(P, λ) above and are used to define weighted characters [A8] :

J_{L}(π, f) = trace (R_{L}(P, π) I_{P} (π, f)) .

The logarithmic derivatives of normalizing factors n_{L}(P, π_{λ}) will also be needed in a relative situation and we shall use a splitting formula:

M_{M}(P, π_{λ}) = ∑_{L∈L(M)} n_{M}^{L}(P, π_{λ})R_{L}(P, π_{λ}) .

V.3 – The spectral term

For sufficiently large T the cutoff trace can be shown [A1b] to be equal (at least for connected groups G) to the trace of the operator induced by the operator R(f) in the image of a projector Λ^{T} called the truncation operator:

J^{T} (f) = trace Λ^{T} R(f)Λ^{T} = ∫_{G(F)\G(𝔸_{F})} Λ^{T} K(x, x) dx

where

Λ^{T} K(x, y) = ∑_{P ⊃P_{0}}(−1)^{a_{P}^{G}} ∑_{δ∈P (F)\G(F)} τ̂_{P} (H_{P} (δx) − T) ∫_{N_{P} (F)\N_{P} (𝔸_{F})} K(nδx, y) dn .

We shall now always assume f left and right K-finite. Using the spectral decomposition and the scalar product formula for truncated Eisenstein series one gets [A4] an explicit formula for J(f). The spectral expansion of J(f) is the sum

– over real positive numbers t ,

– over chains of Levi subgroups M ⊂ L ∈ L ,

– over classes modulo W^{M} of Weyl group elements s ∈ W^{L} such that s(𝔞_{M}) = 𝔞_{M} and without lines of fixed vectors in 𝔞_{M}^{L} ,

– over π in a maximal family of orthogonal irreducible subspace in L^{2}(M(F)\M(𝔸_{F})^{1}) whose infinitesimal character has an imaginary part of modulus t and such that s(π) ≃ π , of the following terms:

\frac{w^{M}}{w^{G}} | det (s − 1)_{𝔞_{M}^{L}} |^{−1} ∫_{i𝔞_{L}^{∗}/𝔞_{G}^{∗}} trace (M_{L}(P, λ)M_{P |P} (s, 0)I_{P} (π_{λ}, f)) dλ

where P is any parabolic subgroup with Levi subgroup M .

One can probably omit the partial summation indexed by the real parameter t. A first step is the trace class property for the discrete spectrum established by W. Müller [M]. But the uniform estimates for the logarithmic derivatives of intertwinning operators that seem necessary to prove the absolute convergence are still missing.

The ”discrete” part J_{disc,t}(f) of the spectral expansion of J(f) is the sum over M, s, π as above of terms with L = G :

\frac{w^{M}}{w^{G}} | det (s − 1)_{𝔞_{M}^{G}} |^{−1} trace (M_{P |P} (s, 0)I_{P} (π_{λ}, f)) .

This contains in particular the contribution of representations that occur discretely in L^{2}(G(F)\G(𝔸_{F})^{1}) : they correspond to terms such that M = G . One can write J_{disc,t}(f) as a sum of characters of irreducible representations of G(𝔸_{F})^{1} and we shall denote by Π_{disc}(G, t) the subset of Π(G) that contributes non trivially:

J_{disc,t}(f) = ∑_{Π_{disc}(G,t)} a_{disc}^{G}(π) trace π(f) .

When π occurs in the discrete automorphic spectrum for G then a_{disc}^{G}(π) is simply its multiplicity. Given π ∈ Π_{disc}(M, t) and λ ∈ i𝔞_{M}^{∗}/𝔞_{L}^{∗} we define

a^{L}(π_{λ}) = a_{disc}^{M}(π)n_{M}^{L}(π_{λ}) .

The above expression for J(f) can now be rewritten [A9b] with the help of weighted characters and of the splitting formula of §6 :

J(f) = ∑_{t} ∑_{L∈L} \frac{w^{L}}{w^{G}} ∫_{Π(L,t)} a^{L}(π) J_{L}(π, f) dπ

the measure (Π(L, t), dπ) is defined by

∫_{Π(L,t)} ϕ(π) dπ = ∑_{M⊂L} \frac{w^{M}}{w^{L}} ∑_{σ∈Π_{disc}(M,t)} ∫_{i𝔞_{M}^{∗} /𝔞_{L}^{∗}} ϕ(σ_{λ}) dλ

The non invariant trace formula is the equality of the geometric and the spectral expansions of J(f) described above:

V.1. Theorem. – For S large enough

J(f) = ∑_{L∈L} \frac{w^{L}}{w^{G}} ∑_{γ∈\{L(F)\}_{L,S}} a^{L}(S, γ) J_{L}(S, γ, f) = ∑_{t} ∑_{L∈L} \frac{w^{L}}{w^{G}} ∫_{Π(L,t)} a^{L}(π) J_{L}(π, f) dπ .

VI – The invariant trace formula

VI.1 – Invariant form

We shall consider below global or S-objects using the same notations. When L ≠ G the distributions J_{L}(∗, f) are non invariant; more precisely if f is K-central one has

J_{L}(∗, f^{y}) = ∑_{Q∈F(L)} \frac{w^{Q}}{w^{G}} J_{L}^{Q}(∗, f_{Q}) u_{Q}(y)

where F(L) is the set of parabolic subgroups Q of G with Levi subgroup M_{Q} containing L .

The key to the invariant form for the trace formula is to assume that f is left and right K-finite and to show, thanks to scalar Paley-Wiener theorems ([CD], [BDK], [R1], etc,) that there exist functions ϕ_{L}(f) in C^{∞}(L(F_{S})) with compact support modulo the center of L such that

trace π(ϕ_{L}(f)) = J_{L}(π, f)

for π ∈ Π(L) tempered up to twist by quasi-characters; the functions ϕ_{L}(f) are defined up to addition of a function with vanishing scalar Fourier transform; of course we may and shall take ϕ_{G}(f) = f . Invariant distributions I_{∗} ”supported on characters” – i.e. such that I_{∗}(f) = 0 whenever trace π(f) = 0 for all π ∈ Π_{temp}(G) – corresponding to the various non invariant distributions J_{∗}, are then uniquely defined, inductively on L, by the following requirements:

J_{M}(∗, f) = ∑_{L∈L(M)} I_{M}^{L}(∗, ϕ_{L}(f))

the induction begins by taking I_{M}^{M} = J_{M}^{M} ; of course this is consistent with the anisotropic case. The existence proof for ϕ_{L} and the I_{∗} uses an intricate local-global inductive argument [A9]. The scalar Paley-Wiener theorem for real groups has been established for connected groups [CD] and in the base change situation [D2] hence the proof is complete only in these cases.

The following lemma is easy but important for applications:

VI.1. Lemma. – If everything is unramified at v and f_{v} is in the spherical Hecke algebra then I_{M}(γ, f_{v}) = J_{M}(γ, f_{v}) .

To see this one may use that R(π) which is a logarithmic derivative of normalized intertwining operators vanishes on the K_{v}-fixed vectors; since we assumed everything unramified and f spherical then ϕ_{L}(f_{v}) can be taken to be zero if L ≠ G .

In the same vein one has the following

VI.2. Lemma. – If π is tempered I_{L}(π, f) = 0 if L ≠ G .

The existence of an invariant form of the trace formula is a formal consequence of the existence of distributions I_{∗} . The geometric and spectral terms have the same expression as in the non invariant form except that the I_{∗} replace the J_{∗} :

VI.3. Theorem. –

I(f) = ∑_{L∈L} \frac{w^{L}}{w^{G}} ∑_{γ∈\{L(F)\}_{L}} a^{L}(γ) I_{L}(γ, f) = ∑_{t} ∑_{L∈L} \frac{w^{L}}{w^{G}} ∫_{Π(L,t)} a^{L}(π) I_{L}(π, f) dπ .

VI.2 – The trace of Hecke operators

The invariant form of the trace formula is well suited to a classical problem: to compute the trace of Hecke operators in a space of automorphic forms satisfying given local conditions. This is a fundamental step in the study of the L-functions of Shimura varieties (cf. [Ko]).

For the sake of simplicity assume G semisimple connected and defined over ℚ. Given a finite dimensional irreducible complex representation V of G_{∞} = G(ℝ) one defines a local system on X_{K} = G(ℚ)\G(𝔸)/K by

F_{V} = G(ℚ)\(V × G(𝔸)/K) .

Assume that G_{∞} is such that rk G_{∞} = rk K_{∞} then the L^{2}-cohomology H_{(2)}^{∗}(X_{K}, F_{V}) is finite dimensional [BC] and can be shown to be equal to the relative Lie algebra cohomology of the space of smooth vectors in the discrete spectrum of the right regular representation of G(𝔸) in L^{2}(G(ℚ)\G(𝔸)) :

H_{(2)}^{∗}(X_{K}, F_{V}) = H^{∗}(𝔤, K_{∞}, L_{disc}^{2}(G(ℚ)\G(𝔸))^{∞} ⊗ V) .

Consider h ∈ C_{c}^{∞}(G(𝔸_{fin})) a smooth compactly supported function on the group of finite adeles and let Λ(h, V) be the alternating sum of the trace of the operator h^{∗} induced in the L^{2}-cohomology:

Λ(h, V) = ∑(−1)^{i} trace (h^{∗}|H_{(2)}^{i}(X_{K}, F_{V})) .

The above equality between cohomology groups shows that

Λ(h, V) = ∑(−1)^{i} trace (h^{∗}|H^{i}(𝔤, K_{∞}, L_{disc}^{2}(G(ℚ)\G(𝔸)^{1})^{∞} ⊗ V)) .

It can be shown to have a nice ”geometric” expression using the trace formula. First it is not difficult (for example using [CD]) to prove that there exist Euler-Poincaré functions i.e. functions f_{V} ∈ C_{c}^{∞}(G(ℝ)) such that

trace π(f_{V}) = ∑(−1)^{i} dim H^{i}(𝔤, K_{∞}, π ⊗ V)

and hence Λ(h, V) equals trace R_{disc}(f_{V} ⊗ h) the trace of the convolution operator defined by f_{V} ⊗ h in the discrete spectrum. Since f_{V} has zero trace in generalized principal series, the spectral term in the invariant trace formula is very simple, in fact:

I(f_{V} ⊗ h) = trace R_{disc}(f_{V} ⊗ h)

The invariant trace formula tells us that for S = {∞} ∪ S_{fin} large enough

Λ(h, V) = ∑_{M∈L} \frac{w^{M}}{w^{G}} ∑_{γ∈\{M(F)\}_{M,S}} a^{M}(S, γ) I_{M}(S, γ, f_{V} ⊗ h)

Then, using splitting formulas and the vanishing of real orbital integrals for non elliptic elements, we may seperate the contributions of the finite adeles and of the reals:

I_{M}(S, γ, f_{V} ⊗ h) = I_{M}(γ, f_{V}) I_{M}^{M}(S_{fin}, γ, h_{M})

Over the finite adeles the contribution of γ is an orbital integral integral on M(𝔸_{fin}) : when γ is semisimple

I_{M}^{M}(S_{fin}, γ, h_{M}) = |D^{M}(S_{fin}, γ)|^{1/2} h(M, γ)

where

h(M, γ) = ∫_{M_{γ}^{0}(𝔸_{fin})\M(𝔸_{fin})} h_{Q}(m^{−1}γm) dṁ

with h_{Q} deduced from h as explained in §3 but with integration over the finite adeles only, and Q is any parabolic subgroup with Levi subgroup M .

The contribution of the real place I_{M}(γ, f_{V}) after some hard work can be shown to be particularly simple since f_{V} is ”discrete” and ”stable”, in fact up to a sign f_{V} is the sum of pseudo-coefficients f_{π_{∞}}, where π_{∞} runs over the L-packet of discrete series attached to V . It can be shown ([A9b], [A10]) that I_{M}(γ, f_{V}) is zero unless γ is an M(ℝ)-elliptic element; for such a γ it can be computed explicitly using Θ_{V} the sum of the characters for V -discrete series. For regular γ define

Φ_{M}(γ, V) = (−1)^{q(G)}|D_{M}^{G}({∞}, γ)|^{1/2}Θ_{V} (γ)

where as usual q(G) = dim G_{∞}^{1}/K_{∞} . These functions can be continuously continued to all M(ℝ)-elliptic elements.

Now introduce the real form compact modulo the center \overline{M_{γ}^{0}} of the connected centralizer, and define a variant of the Tamagawa numbers:

χ(M_{γ}) = (−1)^{q(M_{γ})}d(M_{γ}) \frac{vol (A_{M}M_{γ}^{0}(ℚ)\M_{γ}^{0}(𝔸))}{vol (A_{M}\\overline{M_{γ}^{0}}(ℝ))}

where d(M_{γ}) is the cardinal of L-packets of discrete series on M_{γ}^{0} .

We can now state the final formula [A10]:

Λ(h, V) = ∑_{M∈L}(−1)^{a_{M}^{G}} \frac{w^{M}}{w^{G}} ∑_{γ∈\{M(ℚ)_{ℝ−ell}\}} i^{M}(γ)^{−1}χ(M_{γ})Φ_{M}(γ, V)h_{M,γ} .

VII – Stabilization

The trace formula in its invariant form, given in the previous section, is an identity of sums of distribution that are invariant under conjugacy. But, in the process of comparison of trace formula on different groups a cruder equivalence is relevant: stable conjugacy. Two strongly regular semi-simple elements are said to be stably conjugate if they are conjugate over the algebraic closure.

Over a local field, stable orbital integrals S_{G}(γ, f) of strongly regular semi-simple elements are defined as follows:

S_{G}(γ, f) = ∑_{γ′} I_{G}(γ′, f)

where γ′ runs over a family of representatives of elements that are stably conjugate to γ. A distribution D is said to be stably invariant if D(f) = 0 whenever S_{G}(γ, f) = 0 for all strongly regular semi-simple elements.

The stabilization is the rewriting each side of the trace formula as a sum of distribution that are stably invariant locally everywhere; but since the distribution I(f) itself is not stably invariant, there are error terms. The miracle is that that the error terms can also be written as a linear combination of stable trace formula on a family of groups called endoscopic groups. More precisely, one can define inductively, on any quasi-split groups G′, distributions S^{G′} such that

I^{G}(f) = ∑ ι(G, G′) S^{G′}(f^{G′})

the sum being over endoscopic groups G′ for G. The coefficients ι(G, G′) are rational numbers explicitly given in term of Tamagawa numbers. The functions f^{G′} are the transfer of the function f; their stable orbital integrals can be computed from the orbital integrals of f. This is the existence of the f^{G′} and the compatibility of this transfer with Langlands functoriality at unramified places which is implied by the so-called fundamental lemma. Moreover, if G is itself quasi-split, one can define numbers b^{L} and stable distributions S_{L}(∗, f) such that

S^{G}(f) = ∑_{L∈L} \frac{w^{L}}{w^{G}} ∑_{γ∈\{L(F)\}_{L}} b^{L}(γ) S_{L}(γ, f) = ∑_{t} ∑_{L∈L} \frac{w^{L}}{w^{G}} ∫_{Π(L,t)} b^{L}(π) S_{L}(π, f) dπ .

Stabilization for ordinary endoscopy was established first for groups of invertible elements in quaternion algebras by Jacquet-Langlands [JL], and there the only endoscopic group is GL(2).

Then stabilization was established for SL(2) in [LL] in which case endocopic groups are SL(2) itself and all tori. The next case was U(3) established in [R2]. Under the assumption that the fundamental lemma holds, this is now proved for arbitrary G [A15].

If one works more generally with the twisted trace formula the stabilization yields base change and some other functorialities.

VIII – Appendix 1: Index functions

This appendix reproduces the beginning of [L3] where Index functions are constructed:

VIII.1 – Les représentations spinorielles

Soit G un groupe réductif réel connexe (c’est-à-dire un revêtement fini de la composante neutre du groupe des points réels d’un groupe algébrique réductif). Soit K un sous-groupe compact maximal, θ l’involution de Cartan et 𝔤 = 𝔨 + 𝔭 la décomposition associée. On munit 𝔭 de la métrique induite par la forme de Killing.

Nous supposerons que G que rang G = rang K; en particulier 𝔭 est de dimension paire. On pose q = \frac{1}{2}dim 𝔭. Choisissons un sous-groupe de Cartan compact T de G inclus dans K, et un ordre sur les racines de T dans 𝔤_{ℂ}; on note P l’ensemble des racines positives, P_{K} le sousensemble des racines compactes et P_{n} le sous-ensemble des racines positives non compactes. Comme d’habitude ρ_{P} désigne la demi-somme des racines positives et ρ_{K} la demi-somme des racines positives compactes; ce sont des poids d’un revêtement T̃ de T. Enfin nous noterons W_{K} et W les groupes de Weyl du système de racines de T dans K et 𝔤_{ℂ} respectivement. Si t = exp_{T} H avec H ∈ 𝔱 = Lie(T) on pose

D_{K}(t) = e^{ρ_{K}(H)} ∏_{α∈P_{K}}(1 − e^{−α(H)})

et

D_{P} (t) = e^{ρ_{P} (H)} ∏_{α∈P}(1 − e^{−α(H)}).

On note K̃ le produit fibré de K et de Spin(𝔭) au dessus de SO(𝔭):

K̃ ⟶ Spin(𝔭)
↓  ↓
K ⟶^{Ad} SO(𝔭)

c’est une extension centrale de K par ℤ_{2}. Les représentations spinorielles de K sont, par définition, les représentations de dimension finie de K̃ dont la restriction à ℤ_{2} est un multiple du caractère non trivial. Si λ est un poids d’une représentation spinorielle, alors λ + ρ_{P} − ρ_{K} est un poids de T. Nous noterons R_{spin}(K) le ℤ-module des représentations spinorielles virtuelles.

Au choix du sous-ensemble des racines non compactes positives P_{n} est associé une décomposition 𝔭_{ℂ} = 𝔭^{+} ⊕ 𝔭^{−}. La représentation naturelle de Spin(𝔭) dans S = ∧𝔭^{+} est munie d’une graduation:

S = S^{+} ⊕ S^{−} = (⨁_{i} ∧^{2i}𝔭^{+}) ⊕ (⨁_{i} ∧^{2i+1}𝔭^{+}).

Nous noterons encore S^{±} les représentations de K̃ qui s’en déduisent; ce sont les représentations spinorielles canoniques de K. Le caractère de la différence de ces représentations est donné par

trace S^{+}(t) − trace S^{−}(t) = ∏_{α∈P_{n}}(e^{α(H)/2} − e^{−α(H)/2}) = \frac{D_{P} (t)}{D_{K}(t)}

pour t = exp_{K} H avec H ∈ 𝔱 (cf. [P] ou [AS] (4.13) p.437). On en déduit la

Proposition 1: On a l’équivalence suivante entre représentations virtuelles de K:

(−1)^{q}([S^{+}] − [S^{−}]) ⊗ ([S^{+}] − [S^{−}]) ≃ ∑(−1)^{i}[∧^{i}𝔭_{ℂ}].

VIII.2 – L’indice des opérateurs de Dirac

Soit ξ ∈ R_{spin}(K) une représentation spinorielle virtuelle de K, le produit tensoriel par la différence des représentations spinorielles canoniques

ξ ⊗ ([S^{+}] − [S^{−}])

est une représentation virtuelle (ordinaire) de K; on notera ϕ_{P,ξ} (ou simplement ϕ_{ξ}) la distribution sur G à support dans K produit du caractère virtuel

χ_{P,ξ} = trace (ξ ⊗ ([S^{+}] − [S^{−}]))

par la mesure de Haar normalisée de K.

On note \overline{x} le complexe conjugué de x. Soit π un 𝔤-K-module admissible, l’opérateur

π(\overline{ϕ_{P,ξ}}) = ∫_{K} π(k) \overline{χ_{P,ξ}(k)} dk

est de rang fini et en particulier à trace; on notera I(π; P, ξ) sa trace. Si ξ est effective on a I(π; P, ξ) = trace π(\overline{ϕ_{P,ξ}}) = dim Hom_{K}(ξ ⊗ S^{+}, π) − dim Hom_{K}(ξ ⊗ S^{−}, π).

Par ailleurs il est classique d’associer à une représentation spinorielle effective ξ un opérateur de Dirac:

∇_{P,ξ}^{+} := ∑ X_{i} ⊗ Id_{ξ} ⊗ c(X_{i}) ∈ U(𝔤) ⊗ Hom(ξ ⊗ S^{+}, ξ ⊗ S^{−})

où X_{i} parcourt une base othonormale de 𝔭 et où c(X) est la multiplication de Clifford par X.

Remarque: Cet opérateur définit un élément de K-theorie [∇_{P,ξ}] ∈ K_{0}(C_{r}^{∗}(G)). On peut montrer [Was] que l’application ξ ↦ [∇_{P,ξ}] induit un isomorphisme:

R_{spin}(K) → K_{0}(C_{r}^{∗}(G)).

Soit π une représentation unitaire irréductible, et π̌ sa contragrédiente; on lui associe l’opérateur

π(∇_{P,ξ}^{+}) := ∑ π(X_{i}) ⊗ Id_{ξ} ⊗ c(X_{i}) ∈ End (π) ⊗ Hom(ξ ⊗ S^{+}, ξ ⊗ S^{−}).

On sait (cf. [AS] [CM] [BM]) que I(π̌; P, ξ) est l’indice de π(∇_{P,ξ}^{+}):

I(π̌; P, ξ) = trace π(ϕ_{P,ξ}) = dim Ker π(∇_{P,ξ}^{+}) − dim Coker π(∇_{P,ξ}^{+}).

Si 𝔥 est une sous-algèbre de Cartan de 𝔤 on note j_{𝔥} : Z(𝔤_{ℂ}) → S(𝔥_{ℂ})^{W} l’isomorphisme d’Harish-Chandra.

Proposition 2: Supposons π et ξ irréductibles et soit λ_{ξ} le poids P_{K}-dominant de ξ. L’indice I(π; P, ξ) est non nul seulement si j_{𝔱}^{∗}(λ_{ξ} + ρ_{K}) est le caractère infinitésimal de π.

Preuve: C’est l’assertion (4.20) p.334 de [AS].

□

Si la représentation π est de longueur finie, on dispose d’une expression analytique pour cet indice:

I(π; P, ξ) = ∫_{K} Θ_{π}(k) \overline{χ_{P,ξ}(k)} dk

où Θ_{π} est la fonction analytique sur l’ouvert G′ des éléments réguliers de G définissant le caractère de π (cf. [W] §8.1 et 8.2). Si π est induite à partir d’un sous-groupe parabolique, la fonction Θ_{π} restreinte à K ∩ G′ est nulle, en particulier I(π; P, ξ) = 0. Une propriété de cuspidalité plus fine sera établie au §4. On en déduit la

Proposition 3: Si π est irréductible tempérée et si I(π; P, ξ) ≠ 0, alors π est fondamentale.

Preuve: Soit π irréductible et tempérée; si π n’est pas une série discrète ou une limite de série discrète, c’est-à-dire si π n’est pas fondamentale, il résulte de la classification des représentations tempérées (établie par Knapp et Zuckerman [KZ] pour les groupes linéaires; voir [V, §2] pour le cas général) que π est induite à partir d’un sous-groupe parabolique.

□

Par régularisation au moyen des multiplicateurs d’Arthur-Delorme ([A5], [D1]) (resp. du noyau de la chaleur [BM]) on transforme la mesure en une fonction lisse et à support compact (resp. a decroissance rapide) sur G. Ce sont les ‘fonctions indice’ cherchées... see [L3].

IX – Appendix 2: Adeles

Let F be a number field i.e. a finite extension of ℚ. Consider first the case F = ℚ. One introduce compact rings

\overline{ℤ} = lim_{⟵} ℤ/nℤ

the profinite completion of the integers and, for any prime number p,

ℤ_{p} = lim_{⟵} ℤ/p^{m}ℤ

the ring of p-adic integers. The chinese remainder theorem gives the equality

\overline{ℤ} = ∏_{p} ℤ_{p} .

The ring of finite adeles of ℚ is the tensor product

𝔸_{f} = ℚ ⊗_{ℤ} \overline{ℤ} = lim_{⟶}_{m} \frac{1}{m}\overline{ℤ} = lim_{⟵}_{n} ℚ/nℤ

and the ring of adeles is the product

𝔸 = ℝ × 𝔸_{f} = ℚ ⊗_{ℤ} (ℝ × \overline{ℤ}) .

This is a locally compact commutative ring; it is the “restricted product” of its local completions ℚ_{p} when p is a prime number or ℚ_{∞} = ℝ, with respect to the open compact subrings ℤ_{p} of p-adic integers i.e.:

𝔸 = lim_{⟶}_{S} 𝔸_{S}  with  𝔸_{S} = ℝ × ∏_{p∈S_{f}} ℚ_{p} ∏_{p∉S} ℤ_{p}

where S = {∞} ∪ S_{f} and S_{f} runs over finite set of prime numbers.

Now return to the general case of an arbitrary number field F. The ring of “adeles” of F is the tensor product

𝔸_{F} = F ⊗_{ℚ} 𝔸 = F ⊗_{ℤ} (ℝ × \overline{ℤ}) .

This is a locally compact commutative ring; it is the restricted product of its local completions with respect to the open compact subrings O_{v} of integers at finite places v:

𝔸_{F} = lim_{⟶}_{S} ∏_{v∈S} F_{v} ∏_{v∉S} O_{v} .

The inductive limit is over finite sets S of places of F containing all archimedean places.

One of the basic properties of 𝔸_{F} is that F injects in 𝔸_{F} with a discrete and cocompact image. In fact 𝔸_{F} /F is the Pontryagin dual of the discrete group F. More precisely, the exact sequence

0 → F → 𝔸_{F} → 𝔸_{F} /F → 0

is Pontryagin-auto-dual. The (non-canonical) isomorphism between 𝔸_{F} and its Pontryagin dual 𝔸̂_{F} is defined via the choice of a non trivial (additive) character

ψ : 𝔸_{F} → ℂ^{×}

trivial on F; in fact any x ∈ 𝔸_{F} defines a character ψ_{x} ∈ 𝔸̂_{F}

ψ_{x} : y ↦ ψ(xy)

and one can show that the map x ↦ ψ_{x} is an isomorphism which sends F onto its orthogonal. That 𝔸_{f} is self dual follows from the following isomorphism via Pontryagin duality:

ˆ\overline{ℤ} ≃ lim_{⟶} \frac{1}{n}ℤ/ℤ = ℚ/ℤ .

To see that 𝔸 is self dual we recall that ℝ is self dual.

The multiplicative group scheme 𝔾_{m} is the affine algebraic group scheme defined by the equation xy = 1. The multiplicative group

𝔸_{F}^{×} = 𝔾_{m}(𝔸_{F})

of the adele ring is called the idele group. The idele group is the set of adele points of this algebraic group. Its topology is induced by the inclusion 𝔸_{F}^{×} ⊂ (𝔸_{F})^{2}. This shows that ℚ^{×} injects as a discrete subgroup of 𝔸_{F}^{×}. But the quotient is not compact.

For any place v of F we have an abolute value | |_{v}. Consider x = (x_{v}) ∈ 𝔸_{F}^{×}, then for all but a finite number of places one has |x_{v}|_{v} = 1 hence we may define the absolute value of an idele as the product over all places of the local absolute values:

|x| = ∏_{v} |x_{v}|_{v}

and the map x → log |x| defines a surjective homomorphism

𝔸_{F}^{×} → ℝ .

We denote by 𝔸_{F}^{1} the kernel of the absolute value. The product formula tells us that F^{×} is a (discrete) subgroup of 𝔸_{F}^{1} .

IX.1. Theorem. – The quotient 𝔸_{F}^{1} /F^{×} is compact.

Proof: We prove it for F = ℚ. In fact it is easy to see that

𝔸^{×} = ℚ^{×}. (ℝ^{×} × \overline{ℤ}^{×})

and hence the map

\overline{ℤ}^{×} → 𝔸^{1}/ℚ^{×}

is bijective. For the general case the reader is refered to the literature.

□

Similarly one defines GL(n, 𝔸_{F})^{1} as the kernel of the map g ↦ | det (g)| and more generally, if G is a reductive connected algebraic group G(𝔸_{F})^{1} is the intersection of the kernels of the maps

g ∈ G(𝔸_{F}) ↦ |χ(g)|

where χ is a rational character of G: χ ∈ H^{0}(Gal(\overline{F} /F), X(G)) where X(G) = Hom(G, 𝔾_{m}). Reduction theory gives the

IX.2. Theorem. – The quotient G(F)\G(𝔸_{F})^{1} has finite volume; it is compact if and only if G(F) contains no unipotent element.

To establish the Selberg trace formula for G = SL(2, ℝ) and Γ = SL(2, ℤ) it turn out to be simpler, in particular to get the trace of Hecke operators, to work in the adelic setting with G = GL(2, 𝔸) and Γ = GL(2, ℚ) using that

SL(2, ℤ)\SL(2, ℝ) ≃ GL(2, ℤ)\GL(2, ℝ)^{1} ≃ GL(2, ℚ)\GL(2, 𝔸)^{1}/GL(2, \overline{ℤ})

To see this one observes that

GL(1, ℚ)\GL(1, 𝔸)^{1}/GL(1, \overline{ℤ}) ≃ 1

and

SL(2, ℤ)\SL(2, ℝ) ≃ SL(2, ℚ)\SL(2, 𝔸)/SL(2, \overline{ℤ}) .

40 J.-P. Labesse

X – Bibliography for the notes

[A1a] J. Arthur : A trace formula for reductive groups I, Duke Math. J. 45 (1978), 911-952.

[A1b] J. Arthur : A trace formula for reductive groups II, Comp. Math. 40 (1980), 87-121.

[A2] J. Arthur : Eisenstein series and the trace formula, Proc. Sympos. Pure Math. 33 vol 1 Amer Math. Soc (1979), 253-274.

[A3] J. Arthur : The trace formula for reductive groups, Public. Univ. Paris VII 15 (1979).

[A4a] J. Arthur : On a family of distributions obtained from Eisenstein series I , Amer. J. Math. 104 (1982), 1243-1288.

[A4b] J. Arthur : On a family of distributions obtained from Eisenstein series II, Amer. J. Math. 104 (1982), 1289-1336.

[A5] J. Arthur, A Paley-Wiener theorem for real reductive groups, Acta Math. 150 (1983), 1-89.

[A6] J. Arthur : The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223-293.

[A7] J. Arthur : On a family of distributions obtained from orbits, Canad. J. Math.38 (1986), 179-214 .

[A8a] J. Arthur : Intertwining operators and residues. I. Weighted characters. J. Funct. Anal. 84 (1989), no. 1, 19–84

[A8b] J. Arthur : Intertwining operators and residues. II. Invariant distributions. Compositio Math. 70 (1989), no. 1, 51–99.

[A9a] J. Arthur : The invariant trace formula. I. Local theory. J. Amer. Math. Soc. 1 (1988), no. 2, 323–383.

[A9b] J. Arthur : The invariant trace formula. II. Global theory. J. Amer. Math. Soc. 1 (1988), no. 3, 501–554.

[A10] J. Arthur : The L^{2}-Lefschetz numbers of Hecke operators. Invent. Math. 97 (1989), no. 2, 257–290.

[A11] J. Arthur : Unipotent automorphic representations: global motivation. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 1–75, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

[A12] J. Arthur : Stability and endoscopy: informal motivation. Representation theory and automorphic forms (Edinburgh, 1996), 433–442, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., Providence, RI, 1997.

[A13] J. Arthur : Canonical normalization of weighted characters and a transfer conjecture. C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 2, 33–52

[A14] J. Arthur : Towards a stable trace formula. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998).

[A15a] J. Arthur : A stable trace formula. I

[A15b] J. Arthur : A stable trace formula. II. Global descent. Invent. Math. 143 (2001), no. 1, 157–220.

[A15c] J. Arthur : A stable trace formula. III

[AS] M. Atiyah W. Schmid, A Geometric Construction of the Discrete Series for Semisimple Lie Groups, Invent. Math. 42 (1977), 1-62.

[BC] A. Borel W. Casselman : L^{2}–cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983).

[BDK] J. Bernstein P. Deligne D. Kazhdan : Trace Paley–Wiener theorem for reductive p-adic groups, preprint.

[BM] D. Barbasch H. Moscovici, L^{2}-index and the Selberg trace formula, J. Funct. Anal. 53 (1983), 151-201.

[CD] L. Clozel P. Delorme : Sur le théoreme de Paley–Wiener invariant pour les groupes de lie réductifs réels, C.R.Acad.Sci.Paris t.300 11 (1985).

[CLL] L. Clozel J.-P. Labesse R. Langlands : Morning seminar on the trace formula, Lecture Notes, Institute for Avanced Study, Princeton.

[D1] P. Delorme, Multipliers for the convolution algebra of left and right K-finite compactly supported smooth functions on a semi-simple Lie group, Invent. Math. 75 (1984), 9-23.

[D2] P. Delorme : Thorme de Paley-Wiener invariant tordu pour le changement de base C/R. (French) Compositio Math. 80 (1991), no. 2, 197–228.

[JL] H. Jacquet R. P. Langlands : Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.

[Ko] R. Kottwitz : Shimura varieties and λ-adic representations. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 161–209, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

[KS] A. Knapp E. Stein : Intertwining operators for semisimple groups II, Inv. Math. 60 (1980).

[KZ] A. Knapp G Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math. 116 (1982), 389-501.

[L1] J.-P. Labesse : La formule des traces d’Arthur–Selberg, Séminaire Bourbaki, exposé 636 Nov. 1984.

[L2] J.-P. Labesse : The present state of the trace formula. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 211–226, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

[L3] J.-P. Labesse : Pseudo-coefficients très cuspidaux et K-thorie. Math. Ann. 291 (1991), no. 4, 607–616.

[LL] J.-P. Labesse R. P. Langlands : L-indistinguishability for SL(2). Canad. J. Math. 31 (1979), no. 4, 726–785.

[M] W. Müller : The trace class conjecture in the theory of automorphic forms. Ann. of Math. (2) 130 (1989), no. 3, 473–529.

[P] R. Parthasarathy, Dirac operators and the discrete series, Ann. of Math. 96 (1972) 1-30.

[R1] J. Rogawski : Trace Paley-Wiener theorem in the twisted case. Trans. Amer. Math. Soc. 309 (1988), no. 1, 215–229.

[R2] J. Rogawski : Automorphic representations of unitary groups in three variables. Annals of Mathematics Studies, 123. Princeton University Press, Princeton, NJ, (1990)

[S1] F. Shahidi : Fourier transforms of intertwining operators and Plancherel measures for GL(n), Am. J. Math. 48 (1984).

[S2] F. Shahidi : A proof of Langlands conjecture on Plancherel measures; Complementary series for p-adic groups, preprint (1989).

[V] D. Vogan, Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141-187.

[W] N. Wallach, Real Reductive Groups I, Academic Press (1988)

[Was] A. Wassermann, Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs, C.R.Acad. Sc. Paris 304 No. 18 (1987), 559-562.

44 J.-P. Labesse

XI – Arthur’s bibliography on MathSciNet

This list of James G. Arthur publications on the trace formula and related topics, is extracted from the MathSciNet data base.

[1] A stable trace formula. II. Global descent. Invent. Math. 143 (2001), no. 1, 157–220.

[2] Endoscopic L-functions and a combinatorial identity. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. Canad. J. Math. 51 (1999), no. 6, 1135–1148.

[3] Stabilization of a family of differential equations. The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), 77–95, Proc. Sympos. Pure Math., 68, Amer. Math. Soc., Providence, RI, 2000.

[4] On the transfer of distributions: weighted orbital integrals. Duke Math. J. 99 (1999), no. 2, 209–283.

[6] L^{2}-cohomology and automorphic representations. Canadian Mathematical Society. 1945–1995, Vol. 3, 1–17, Canadian Math. Soc., Ottawa, ON, 1996.

[7] On local character relations. Selecta Math. (N.S.) 2 (1996), no. 4, 501–579.

[8] Towards a stable trace formula. Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998). Doc. Math. 1998, Extra Vol. II, 507–517 (electronic).

[9] Canonical normalization of weighted characters and a transfer conjecture. C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 2, 33–52.

[10] Stability and endoscopy: informal motivation. Representation theory and automorphic forms (Edinburgh, 1996), 433–442, Proc. Sympos. Pure Math., 61, Amer. Math. Soc., Providence, RI, 1997.

[11] Towards a local trace formula. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), 1–23, Johns Hopkins Univ. Press, Baltimore, MD, 1989.

[12] The trace Paley Wiener theorem for Schwartz functions. Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), 171–180, Contemp. Math., 177, Amer. Math. Soc., Providence, RI, 1994.

[13] On the Fourier transforms of weighted orbital integrals. J. Reine Angew. Math. 452 (1994), 163–217.

[14] On elliptic tempered characters. Acta Math. 171 (1993), no. 1, 73–138.

[16] Some problems in local harmonic analysis. Harmonic analysis on reductive groups (Brunswick, ME, 1989), 57–78, Progr. Math., 101, BirkhSuser Boston, Boston, MA, 1991.

[17] (with Gelbart, Stephen) Lectures on automorphic L-functions. L-functions and arithmetic (Durham, 1989), 1–59, London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, 1991.

[18] A local trace formula. Inst. Hautes Études Sci. Publ. Math. No. 73 (1991), 5–96.

[19] Unipotent automorphic representations: global motivation. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 1–75, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

[20] The L^{2}-Lefschetz numbers of Hecke operators. Invent. Math. 97 (1989), no. 2, 257– 290.

[21] Unipotent automorphic representations: conjectures. Orbites unipotentes et représentations, II. Ast*risque No. 171-172 (1989), 13–71.

[22] Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one. Dissertation, Yale University, New Haven, CT, 1970. Math. Surveys Monogr., 31, Representation theory and harmonic analysis on semisimple Lie groups, 13–100, Amer. Math. Soc., Providence, RI, 1989.

[23] (with Clozel, Laurent) Simple algebras, base change, and the advanced theory of the trace formula. Annals of Mathematics Studies, 120. Princeton University Press, Princeton, NJ, 1989.

[24] Intertwining operators and residues. II. Invariant distributions. Compositio Math. 70 (1989), no. 1, 51–99.

[25] Intertwining operators and residues. I. Weighted characters. J. Funct. Anal. 84 (1989), no. 1, 19–84.

[26] Characters, harmonic analysis, and an L^{2}-Lefschetz formula. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 167–179, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988.

[27] The trace formula and Hecke operators. Number theory, trace formulas and discrete groups (Oslo, 1987), 11–27, Academic Press, Boston, MA, 1989.

[28] The characters of supercuspidal representations as weighted orbital integrals. Proc. Indian Acad. Sci. Math. Sci. 97 (1987), no. 1-3, 3–19 (1988).

[29] The invariant trace formula. II. Global theory. J. Amer. Math. Soc. 1 (1988), no. 3, 501–554.

[30] The local behaviour of weighted orbital integrals. Duke Math. J. 56 (1988), no. 2, 223–293.

[31] The invariant trace formula. I. Local theory. J. Amer. Math. Soc. 1 (1988), no. 2, 323–383.

[32] A measure on the unipotent variety. Canad. J. Math. 37 (1985), no. 6, 1237–1274.

[34] On a family of distributions obtained from orbits. Canad. J. Math. 38 (1986), no. 1, 179–214.

[35] The trace formula for noncompact quotient. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 849–859, PWN, Warsaw, 1984.

[36] On a family of distributions obtained from Eisenstein series. I. Application of the Paley-Wiener theorem. Amer. J. Math. 104 (1982), no. 6, 1243–1288.

[37] On some problems suggested by the trace formula. Lie group representations, II (College Park, Md., 1982/1983), 1–49, Lecture Notes in Math., 1041, Springer, Berlin, 1984.

[38] The trace formula for reductive groups. Conference on automorphic theory (Dijon, 1981), 1–41, Publ. Math. Univ. Paris VII, 15, Univ. Paris VII, Paris, 1983.

[39] Multipliers and a Paley-Wiener theorem for real reductive groups. Representation theory of reductive groups (Park City, Utah, 1982), 1–19, Progr. Math., 40, BirkhSuser Boston, Boston, MA, 1983.

[40] On a family of distributions obtained from Eisenstein series. II. Explicit formulas. Amer. J. Math. 104 (1982), no. 6, 1289–1336.

[41] A Paley-Wiener theorem for real reductive groups. Acta Math. 150 (1983), no. 1-2, 1–89.

[42] Automorphic representations and number theory. 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980), pp. 3–51, CMS Conf. Proc., 1, Amer. Math. Soc., Providence, R.I., 1981.

[43] The trace formula in invariant form. Ann. of Math. (2) 114 (1981), no. 1, 1–74.

[44] On the inner product of truncated Eisenstein series. Duke Math. J. 49 (1982), no. 1, 35–70.

[45] A trace formula for reductive groups. II. Applications of a truncation operator. Compositio Math. 40 (1980), no. 1, 87–121.

[46] Eisenstein series and the trace formula. Automorphic forms, representations and Lfunctions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 253–274, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

[47] Harmonic analysis of invariant distributions. Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977), pp. 384–393, Queen’s Papers in Pure and Appl. Math., 48, Queen’s Univ., Kingston, Ont., 1978.

[48] A trace formula for reductive groups. I. Terms associated to classes in G(Q). Duke Math. J. 45 (1978), no. 4, 911–952.

[49] A truncation process for reductive groups. Bull. Amer. Math. Soc. 83 (1977), no. 4, 748–750.

[50] A theorem on the Schwartz space of a reductive Lie group. Proc. Nat. Acad. Sci. U.S.A. 72 (1975), no. 12, 4718–4719.

[51] The characters of discrete series as orbital integrals. Invent. Math. 32 (1976), no. 3, 205–261.

[52] Some tempered distributions on semisimple groups of real rank one. Ann. of Math.

(2) 100 (1974), 553–584.

[53] The Selberg trace formula for groups of F -rank one. Ann. of Math. (2) 100 (1974), 326–385.