LECTURES ON THE ARTHUR–SELBERG TRACE FORMULA

STEVE GELBART

MSRI, Spring 1995

These are Notes prepared for nine lectures given at the Mathematical Sciences Research Institute, MSRI, Berkeley during the period January–March 1995. It is a pleasant duty to record here my gratitude to MSRI, and its staff, for making possible this 1994–95 Special Year in Automorphic Forms, and for providing such a setting for work.

The purpose of these Notes is to describe the contents of Arthur’s earlier, foundational papers on the trace formula. In keeping with the introductory nature of the lectures, we have sometimes illustrated the ideas of Arthur’s general theory by applying them in detail to the case of GL(2); we have also included a few lectures on the “simple trace formula” (and its applications), and on Jacquet’s relative trace formula.

The TEX preparation of these Notes I owe to Wendy McKay, who patiently and professionally transformed my messy scrawl into something readable; her expertise, and good cheer under the pressure of weekly deadlines, is something I shall not soon forget.

I wish to thank the auditors of these lectures for their interest, and , J. Bernstein, D. Goldberg, E. Lapid, C. Rader, S. Rallis, A. Reznikov, and D. Soudry, for their helpful suggestions and explanations. Finally, I wish to thank J. Arthur, H. Jacquet, and J. Rogawski for many tutorials on these and related topics over the past year; I hope they do not mind seeing some of their comments reappear in these Notes.

Table of Contents

Lecture I. Introduction to the Trace Formula

Lecture II. Arthur’s Modified Kernels I: The Geometric Terms

Lecture III. Arthur’s Modified Kernels II: The Spectral Terms

Lecture IV. More Explicit Forms of the Trace Formula

Lecture V. Simple Forms of the Trace Formula

Lecture VI. Applications of the Trace Formula

Lecture VII. (G, M)-Families and the Spectral J_{χ}(f)

Lecture VIII. Jacquet’s Relative Trace Formula

Lecture IX. Some Applications of Paley–Wiener, and Concluding Remarks

Lecture I. Introduction to the Trace Formula

The ultimate purpose of the trace formula is to better understand the structure of automorphic forms on a general reductive group G. In particular, one wants to understand the decomposition of the right regular representation R_{0}(g) of G(𝔸) in the space of cusp forms L_{0}^{2}(G(F) \ G(𝔸)). The original idea of the (Selberg) trace formula—roughly speaking—was to describe the trace of the integral operator R_{0}(f) = ∫_{G(𝔸)} f(g)R_{0}(g)dg for f in C_{c}^{∞}(G(𝔸)); this Selberg succeeded to do for certain special G. On the other hand, Arthur’s trace formula is valid for arbitrary G, and asserts the equality of two different sums of distributions on G, namely:

∑_{𝔬∈𝔒} J T_{𝔬} (f) = ∑_{χ∈𝔛} J_{χ}^{T} (f). (0.1)

Here the left side of (0.1) is summed over certain equivalence classes in the group of rational points G(F); it is called the geometric side of the trace formula. On the right side of (0.1), the sum is over certain “cuspidal data” χ = {(M, σ)} of G; this is the spectral side of the trace formula (one of whose terms is trR_{0}(f)).

Our purpose in these lectures is to describe and develop this formula in complete detail, but mostly in the context of GL(2). We also wish to describe some refinements of (0.1) necessary for applications to automorphic forms. The basic references are Arthur’s original papers [A1]–[A10]. Earlier expositions of the GL(2) trace formula, such as [Ge] and [GJ], will be referred to occasionally; but as they mostly predate Arthur’s general development of the trace formula, their point of view will largely be ignored, and their arguments not reproduced here.

1. Some history. “The” trace formula was introduced by Selberg, in the context of a semi-simple Lie group G and discrete subgroup Γ, in his famous 1956 paper [Se]. In that paper Selberg first of all described a general formula in the case of compact quotient Γ \ G (see the next section below for a brief description). Secondly, he treated in detail certain special non-compact quotient cases such as SL_{2}(ℤ) \ SL_{2}(ℝ); here Selberg analytically continued Eisenstein Series E(z, s) in order to handle the continuous spectrum of L^{2}(Γ \ G), and derived a formula for the trace of R(f) in the space of cusp forms. In both the case of compact and non-compact quotient, Selberg was motivated by applications to geometry and number theory (lengths of geodesics in the Riemann surface in the case of compact quotient, traces of Hecke operators in case of non-compact quotient, etc.). But despite the obvious desirability of generalizing these results in the case of non-compact quotient, the obstacles—such as analytically continuing the Eisenstein series in higher rank—were formidable. A big breakthrough came in the 1960’s when Langlands developed a theory of Eisenstein series valid for any reductive group G, and used it to describe the continuous spectrum of L^{2}(G(F) \ G(𝔸)) (cf. [La 1]). As an outgrowth of this theory, Langlands was led to his amazing conjectures about automorphic forms and their L-functions, i.e., the “Langlands program”, with its cornerstone the principle of functoriality of automorphic forms. A typical result in this program is the following:

Theorem 1 ([JL]). Let D be a division quaternion algebra over a number field F and G′ = D^{∗}. Then to each automorphic cuspidal representation σ of G′ (i.e., an irreducible G′(𝔸)-submodule of L^{2}(Z(𝔸)G′(F)\G′(𝔸)) of dimension greater than one) there exists a corresponding automorphic cuspidal representation π = π(σ) of G(𝔸) = GL_{2}(𝔸) (an irreducible G(𝔸)-submodule of the space of cusp forms L_{0}^{2}(Z(𝔸)G(F) \ G(𝔸)) with the property that for each place v in F unramified for D (i.e. where G′(F_{v}) ≃ GL_{2}(F_{v})), σ_{v} ≃ π_{v}.

Interestingly enough, this correspondence (the well-known Jacquet–Langlands correspondence) can be proved using any one of the three basic tools of the theory of automorphic forms: the theory of automorphic L-functions, the theory of theta-series liftings, or the trace formula (for both G′ and G = GL_{2}, essentially as developed by Selberg). However, in this case at least, the trace formula approach gives the strongest payoff, namely a characterization of the image of the correspondence σ → π(σ): a cuspidal representation π of G(𝔸) is of the form π(σ) for some σ on a given G′_{D} if and only if for each ramified prime v, π_{v} is a square integrable (discrete series) representation of G_{v} = GL_{2}(F_{v}).

Subsequently, a (“twisted”) trace formula for GL_{2} was developed by Saito-Shintani and Langlands to prove another important instance of functoriality, namely base change (with spectacular applications to Artin’s conjecture for Galois representations, and hence ultimately to Wiles’ proof of Fermat …). Thus the need to generalize the trace formula to a general reductive group G was clear, and this is precisely what Arthur did in his papers between the early 1970’s and late 1980’s.

2. The Case of Compact Quotient. In order to motivate Arthur’s work, let us explain how the form (2) takes in the case of compact quotient. Thus G is a semisimple group over (the number field) F, with G(F)\G(𝔸) compact. In particular, G is anisotropic, without unipotent elements. With R(g) the right regular representation in L^{2}(G(F) \ G(𝔸)) and f in C_{c}^{∞}(G(𝔸)), a straightforward computation gives

(R(f)ϕ)(x) = ∫_{G(𝔸)} f(y)ϕ(xy) dy =

∫_{G(𝔸)} f(x^{−1}y)ϕ(y) dy = ∫_{G(F)\G(𝔸)}( ∑_{γ∈G(F)} f(x^{−1}γy))ϕ(y) dy,

i.e., R(f) is an integral operator on ϕ in L^{2}(G(F) \ G(𝔸)), with kernel

K_{f}(x, y) = ∑_{γ∈G(F)} f(x^{−1}γy).

Since G(F) \ G(𝔸) is compact, it is then a well known theorem (cf. [GGPS], Chapter I), that R(f) is of trace class, and that

trR(f) = ∫_{G(F)\G(𝔸)} K_{f}(x, x) dx,

the integral of the kernel over the “diagonal subgroup” G(F) \ G(𝔸)^{Δ}. In this case, the (Selberg) trace formula amounts to a calculation of traceR(f) using two different expressions for this kernel. On the one hand, we can write

K_{f}(x, y) = ∑_{𝔬∈𝔒} K_{𝔬}(x, y)

where 𝔒 denotes the set of conjugacy classes in G(F), and

K_{𝔬}(x, y) = ∑_{γ∈𝔬={γ_{0}}} f(x^{−1}γy) = ∑_{δ∈G_{γ_{0}}(F)\G(F)} f(x^{−1}δ^{−1}γ_{0}δx)

where G_{γ_{0}}(F) is the centralizer of γ_{0} in G(F). Then

J_{𝔬}(f) = ∫_{G(F)\G(𝔸)} K_{f}(x, x) dx = ∫_{G(F)\G(𝔸)} ∑_{δ∈G_{γ_{0}}(F)\G(F)} f(x^{−1}δ^{−1}γ_{0}δx) dx =

∫_{G_{γ_{0}}(F)\G(𝔸)} f(x^{−1}γ_{0}x) dx = meas(G_{γ_{0}}(F) \ G_{γ_{0}}(𝔸)) ∫_{G_{γ_{0}}(𝔸)\G(𝔸)} f((x^{−1}γ_{0}x) dx,

and thus

trR(f) = ∑_{𝔬∈𝔒} J_{𝔬}(f) = ∑_{{γ}} meas(G_{γ}(F) \ G_{γ}(𝔸)) ∫_{G_{γ}(𝔸)\G(𝔸)} f((x^{−1}γx) dx.

On the other hand, the compactness of G(F) \ G(𝔸) also implies a discrete decomposition R(g) = ∑_{π} m_{π}π(g) where π runs through the irreducible “automorphic” representations of G(𝔸), and m_{π} < ∞ is the multiplicity of π in R. Equivalently, L^{2}(G(F) \ G(𝔸)) = ∑ L_{π} where L_{π} is the invariant subspace realizing the m_{π} copies of π, and R(f) = ∑ R_{π}(f) where each R_{π}(f) is an integral operator in L_{π} with kernel

K_{π}(x, y) = ∑_{ϕ an ON basis forL_{π}}(R(f)ϕ)(x)\overline{ϕ(y)}.

Thus we also have

trR(f) = ∑_{χ∈{(G,π)}} J_{χ}(f),

where

J_{χ}(f) = ∫_{G(F)\G(𝔸)} K_{π}(x, x) dx.

The power of the trace formula derives from the fact that this equality

∑ J_{𝔬}(f) = ∑ J_{χ}(f), (2.1)

namely

∑_{\{γ\}} m_{γ} ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γx) dx = ∑_{π} ∫_{G(F)\G(𝔸)}(∑_{ϕ}(R(f)ϕ)(x)\overline{ϕ(x)}) dx

relates two completely different kinds of data, geometric and spectral (the latter of which we rarely know much about, but very much want to!).

3. Difficulties in the Non-Compact Case. Arthur’s great contribution was to figure out how to overcome all the obstacles that present themselves in generalizing (2.1) from the compact to the non-compact quotient setting. Some of the principal problems in the non-compact setting are these:

(a) Although R(f) is still an integral operator, with kernel

K_{f}(x, y) = ∑_{γ} f(x^{−1}γy),

this kernel is no longer integrable over the diagonal; and

(b) The regular representation R(g) no longer decomposes discretely, Eisenstein series are required to describe the continuous spectrum, and R(f) is of course no longer of trace class. Nevertheless, Arthur derives a formula (0.1) which still relates “geometric” and “spectral” distributions attached to G—and has powerful applications to number theory (even if neither side of (0.1) represents a formula for trR(f)!). For example, in this formula,

J T_{𝔬} (f) = ∫_{Z(𝔸)G(F)\G(𝔸)} KT_{𝔬} (x, x) dx,

where KT_{𝔬} is a modification of the kernel

K_{𝔬}(x, x) = ∑_{γ∈𝔬} f(x^{−1}γx)

which can be integrated over the diagonal. To see why this modification is really needed, and what it accomplishes, let us look at the example of G = GL(2), and 𝔬 the hyperbolic conjugacy class

𝔬 = \{δ^{−1}({α \atop 0} {0 \atop 1})δ : δ ∈ M(F) \ G(F)\},

where M = \{( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )\}. In this case,

J_{𝔬}(f) = ∫_{Z(𝔸)G(F)\G(𝔸)} K_{𝔬}(x, x) dx

(where Z is the center of scalar matrices). A formal computation then shows

J_{𝔬}(f) = ∫_{Z(𝔸)G(F)\G(𝔸)} ∑_{δ∈M(F)\G(F)} f(x^{−1}δ^{−1}({α \atop 0} {0 \atop 1})δx) dx
= ∫_{Z(𝔸)G(F)\G(𝔸)} f(x^{−1}({α \atop 0} {0 \atop 1})x) dx
= ∫_{M(𝔸)\G(𝔸)}(∫_{Z(𝔸)M(F)\M(𝔸)} dy)f(x^{−1}({α \atop 0} {0 \atop 1})x) dx,

i.e.,

J_{𝔬}(f) = m(Z(𝔸)M(F) \ M(𝔸))F_{f}(({α \atop 0} {0 \atop 1})),

with F_{f}(({α \atop 0} {0 \atop 1})) the nice “orbital integral” ∫_{M(𝔸)\G(𝔸)} f(x^{−1}({α \atop 0} {0 \atop 1})x) dx of a compactly supported function f in C_{c}^{∞}(Z(𝔸)\G(𝔸)). Thus J_{𝔬}(f) is infinite for the simple reason that

m(Z(𝔸)M(F) \ M(𝔸)) = m(F^{∗} \ 𝔸^{∗}) = ∞.

To compensate for this, Arthur “modifies” K_{𝔬} to KT_{𝔬}, using a truncation parameter T = (T_{1}, T_{2}) in the Lie algebra of M. As we shall see, this modification operator (in this case) “truncates” F^{∗} \ 𝔸^{∗} to a finite interval of length T_{2} − T_{1}, but turns F_{f} into a “weighted” orbital integral! In general, for an arbitrary G, it will turn out that K_{𝔬} will be integrable “over the diagonal” only when 𝔬 intersects no proper parabolic subgroup P of G. Of course for anisotropic G this always happens, but in general one really needs to modify K_{𝔬} (as in the above example), prove that the modified kernels KT_{𝔬} are indeed integrable, and then evaluate them (as generalized weighted orbital integrals).

What about the spectral side? Here it turns out that L^{2}(Z(𝔸)G(F) \ G(𝔸)) = ∑_{χ∈𝔛} L_{χ}^{2}, where each L_{χ}^{2} is a G(𝔸)-invariant (not necessary discretely decomposable) submodule indexed by certain cuspidal data 𝔛 = {(M, σ)}, where M is a Levi subgroup of a parabolic P ⊂ G, and σ is a cuspidal automorphic representation of M(𝔸). Thus we have K(x, y) = ∑ K_{χ}(x, y), the spectral decomposition of K_{f}(x, y) corresponding to (0.1). Here it also turns out that K_{χ}(x, y) will not be integrable over the diagonal, unless the cuspidal data (M, σ) = (G, π) (i.e. P = G). Thus we again need to deal with modified K_{χ}^{T} (which are integrable …), eventually arriving at the trace formula (0.1) by integrating the equality

∑_{𝔬} KT_{𝔬} (x, y) = ∑_{χ} K_{χ}^{T} (x, y) (3.1)

over the diagonal.

Remarks. (1) The expressions on either side of (3.1) no longer (generally) represent the kernel K_{f}(x, y) = ∑ f(x^{−1}γy), and hence (0.1) is no longer a real trace formula at all. Nevertheless, what is true is the following. Let 𝔛(G) be the subset of cuspidal data χ = (M, σ) ∈ 𝔛 with M = G. Then ∑_{χ∈𝔛(G)} K_{χ}(x, y) is the kernel of R(f) restricted to L_{0}^{2}(Z(𝔸)G(F) \ G(𝔸)), it is integrable over the diagonal, and (0.1) asserts that

trR_{0}(f) = ∑_{χ∈𝔛(G)} J_{χ}(f) = ∑_{𝔬} J T_{𝔬} (f) − ∑_{𝔛\𝔛(G)} J_{χ}^{T} (f), (3.2)

i.e., Arthur’s trace formula may be viewed as a trace formula after all.

(2) In most applications of the trace formula to automorphic forms we want to compare the formula for one group with the trace formula for another. In that case the form of (0.1) is no less useful than (3.2).

4. Possible Applications. We have already mentioned the applications to quaternion algebras and base change, both of which we plan to discuss (at the level of GL(2), not GL(n) - which is the subject matter of [AC], using the more difficult GL(n) trace formulas). We also plan to discuss (i) Jacquet’s relative trace formula, where the idea is to integrate the (different expressions for the) kernel over interesting non-diagonal subgroups of G × G; and (ii) applications of the simple trace formula such as to the embedding of a (local) discrete series representation of GL_{2}(F_{v}) into a (global) cuspidal representation of GL_{2}(𝔸). In all these applications, one needs to further refine Arthur’s first form of the trace formula. This is because (as already hinted) the “truncation” i.e., modification process used to make K_{𝔬} or K_{χ} integrable introduces various defects into the distributions J T_{𝔬} (f) and J_{χ}^{T} (f). For example, they may no longer be invariant distributions (invariant under conjugation by G(𝔸)), or factorizable as the product of local distributions over G(F_{v}). Thus one needs to replace and rearrange the sums of distributions appearing in (0.1) by still more complicated but ultimately more practical expressions.

References

[A1] Arthur, J., “The Selberg trace formula for groups of F -rank one”, Annals of Math. 100 (1974), pp. 236–385.

[A2] Arthur, J., “A Trace formula for reductive groups I: Terms associated to classes in G(ℚ)”, Duke Math. J., Vol. 45, No. 4 (1978), pp. 911–952.

[A3] Arthur, J., “A Trace formula for reductive groups II: Applications of a truncation operator”, Compos. Math. 40 (1980), pp. 87–121.

[A4] Arthur, J., “A trace formula in invariant form”, Annals of Math. 113 (1981), pp. 1–74.

[A5] Arthur, J., “On a family of distributions obtained from Eisenstein series I: Application of the Paley–Wiener theorem”, Amer. J. of Math., Vol. 104, No. 6 (1982), pp. 1243–1288.

[A6] Arthur, J., “On a family of distributions obtained from Eisenstein series II: Explicit formulas”, Amer. J. of Math., Vol. 104, No. 6 (1982), pp. 1289–1336.

[A7] Arthur, J., “The trace formula for reductive groups”, Lecture for Journées Automorphes, Dijon, 1981.

[AC] Arthur, J. and Clozel, L., Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Math. Studies No. 120, Princeton Univ. Press, 1989.

[Ge] Gelbart, S., Automorphic Forms on Adele Groups, Annals of Math. Studies No. 83, Princeton Univ. Press, 1975.

[GJ] Gelbart, S. and Jacquet, H., “Forms of GL(2) from the analytic point of view”, in Proc. Symp. Pure Math., Vol. 33, Part I, pp. 213–251.

[GGPS] Gelfand, I. M., Graev, M., and Piatetski-Shapiro, I., Automorphic Functions and Representation Theory, Saunders, 1969.

[JL] Jacquet, H., and Langlands, R. P., Automorphic Forms on GL(2), Springer Lecture Notes, Vol. 114, 1970.

[La1] Langlands, R. P., “On the functional equations satisfied by Eisenstein series , Springer Lecture Notes, Vol. 544, 1976.

[Se] Selberg, A., “Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces, with applications to Dirichlet series”, J. Indian Math. Soc. 20 (1956), pp. 47-87; see also Proc. Int. Cong. Math. 1962, pp. 177–189.

Lecture II. Arthur’s Modified Kernels I: The Geometric Terms

Our purpose here is to develop the geometric side of Arthur’s first form of the trace formula

∑_{𝔬∈𝔒} J T_{𝔬} (f) = ∑_{χ∈𝔛} J_{χ}^{T} (f),

for the group G = GL(2). In particular, we must introduce the modified kernels kT_{𝔬} (x, f), and prove they are integrable.

N.B. The treatment of the trace formula in [Ge] and [GJ] largely ignored the integrability of these modified kernels.

1. The Definition of K_{𝔬}. In the case of compact quotient, the geometric side of the trace formula is summed over ordinary G(F)-conjugacy classes 𝔬 in G(F) (or in Z(F)\G(F), where Z is the center of G). For arbitrary G, it turns out to be best to make the following definition: γ_{1} and γ_{2} in Z(F)\G(F) are called equivalent, or “conjugate”, if their semi-simple components are G(F) conjugate in the usual sense. That is, γ_{1} ∼ γ_{2} means, with the Jordan decompositions γ_{i} = γ_{i}^{s}γ_{i}^{u} (i = 1, 2), that γ_{1}^{s} is G(F)-conjugate to γ_{2}^{s}. For each such equivalence class 𝔬, and for f ∈ C_{c}^{∞}(Z(𝔸)\G(𝔸)), set

K_{𝔬}(x, y) = ∑_{γ∈𝔬} f(x^{−1}γy),

and let 𝔒 denote the collection of all such classes 𝔬 in Z(𝔸)\G(𝔸).

Note that if γ is elliptic, i.e. if γ is not conjugate in G(F) to an element of any proper parabolic subgroup P (F), then it is automatically semi-simple. (Its eigenvalues do not lie in F, and hence are distinct.) Thus our new notion of conjugacy reduces to ordinary conjugacy.

On the other hand, suppose γ is G(F)-conjugate to an element p in a proper parabolic subgroup P (F). For GL(2), this means P is the Borel subgroup B = \{({a \atop 0} {x \atop b})\}, so modulo Z(F),

p = p^{s}p^{u} with p^{s} = ({α \atop 0} {0 \atop 1}), and p^{u} = ({1 \atop 0} {y \atop 1} ).

Case (a): α = 1. In this case, the class 𝔬 containing γ consists of two ordinary unipotent conjugacy classes in Z(F)\G(F): the trivial class {1}, and the ordinary conjugacy class of any non-trivial unipotent element ({1 \atop 0} {x_{0} \atop 1} ) (since ({α \atop 0} {0 \atop 1}) already con-

jugates ({1 \atop 0} {x_{0} \atop 1} ) to ({1 \atop 0} {αx_{0} \atop 1} ) ).

Case (b): α ≠ 1. In this case, 𝔬 is an ordinary hyperbolic conjugacy class of the form

𝔬_{α} = \{δ^{−1}({α \atop 0} {0 \atop 1})δ | δ ∈ M(F)\G(F)\} .

Note that M(F) is just the centralizer G_{(_{{α \atop 0} {0 \atop 1}})}(F), and that ({α \atop 0} {x \atop 1}) also belongs to 𝔬_{α}

(since ({α \atop 0} {x \atop 1}) = ({1 \atop 0} {y \atop 1} )^{−1}({α \atop 0} {0 \atop 1})({1 \atop 0} {y \atop 1} )^{−1} where y = x/(α − 1) ).

2. The Integrability of Elliptic K_{𝔬}. As suggested in Lecture I, it is a general phenomenon that for an elliptic class 𝔬, the kernel K_{𝔬}(x, y) is integrable over the diagonal subset Z(𝔸)G(F)\G(𝔸). We now prove this for G = GL(2), following the general arguments of [A1]. The crucial step is a lemma which first requires some preliminary notions to be recalled.

According to the Iwasawa decomposition for G, we have

G(𝔸) = N(𝔸) M(𝔸) K, (2.1)

where N is the standard unipotent subgroup \{({1 \atop 0} {y \atop 1} )\}, and K is the standard maximal

compact subgroup ∏ K_{ν} of G(𝔸). If we denote by 𝔞_{M} the Lie algebra of M = \{({a \atop 0} {0 \atop b})\}, i.e. ,

𝔞_{M} = Hom(X(M), ℝ) ≈ ℝ^{2},

then we can define the familiar function

H_{M} : M(𝔸) → 𝔞_{M} by H_{M} ({a \atop 0} {0 \atop b}) = (log |a|_{𝔸}, log |b|_{𝔸}),

and extend it to G(𝔸) by way of the decomposition 2.1, i.e. if g = nak in G(𝔸), then

H_{M}(g) = H_{M}(a).

Lemma 2.1 ( see [A1, p 362] ). Suppose Ω is a subset of G(𝔸) which is compact modulo Z(𝔸). Then there exists a number d_{Ω} > 0 with the following property: if γ ∈ G(F) is such that

g^{−1}γng ∈ Ω

for some n ∈ N(𝔸) and g ∈ G(𝔸) with αH(g) > d_{Ω}, then γ ∈ B(F).

Assuming this Lemma (for the moment), let us quickly dispose of the integrability of K_{𝔬}. For this, and for other purposes later on, we also need to recall the notion of a Siegel domain, together with an explicit integration formula for its volume.

First we refine the Iwasawa decomposition 2.1 by writing, for each g in G(𝔸),

g = znh_{t}mk (2.2)

where z ∈ Z(𝔸), n ∈ N(𝔸), k ∈ K,

h_{t} = (\begin{matrix}e & ^{t} \\ 0 \end{matrix} \begin{matrix}0 \\ e & ^{−t} \end{matrix} ),

with e^{t} an idele equal to 1 at every finite place and the real number e^{t} at each infinite place, and

m = ({a \atop 0} {0 \atop 1}),

with a ∈ 𝔸^{1} = {a ∈ 𝔸^{×} | |a|_{𝔸} = 1}. The corresponding integration formula, for f on Z(𝔸)\G(𝔸), is

∫_{Z(𝔸)\G(𝔸)} f(g) dg = ∫ ∫ ∫ ∫_{−∞}^{∞} f(natk)e^{−2t} dt d^{×}a dn dk. (2.3)

Now for each compact subset C_{1} of N(𝔸), compact subset C_{2} of 𝔸^{1}, and real number c (not necessarily positive), let 𝔖_{c} denote the set of points g in G(𝔸) of the form (2.2) with n ∈ C_{1}, a ∈ C_{2} and t > c/2. Such a set is called a Siegel set. Note that

𝔖_{c} ⊂ { g ∈ G(𝔸) | H(g) > c }.

Fact 1 (see [Go, Thm 9]). For any c_{1}, c_{2},

𝔖_{c_{1}} ∩ γ𝔖_{c_{2}} ≠ ∅

for only finitely many γ in G(F) modulo B(F).

Fact 2 (see [Go, p 16]). If 𝔖_{c} is sufficiently large (i.e. C_{1} and C_{2} are sufficiently large, and c is sufficiently small), then

G(𝔸) = G(F) · 𝔖_{c}. (2.4)

In other words, 𝔖_{c} is essentially a fundamental domain for G(F) acting on G(𝔸); more precisely, Fact 2 says only that certain 𝔖_{c} contain a fundamental domain. Henceforth, we always assume that 𝔖_{c} is so chosen that 2.4 holds, and we refer to such a 𝔖_{c} as a Siegel domain.

Remark. From the above, it is clear that

Z(𝔸)G(F)\G(𝔸)

has finite volume. In fact, by 2.3 and 2.4,

∫_{Z(𝔸)G(F)\G(𝔸)} dg ≤ ∫_{Z(𝔸)\𝔖_{c}} dg = ∫ ∫ ∫ (∫_{c}^{∞} e^{−2t} dt) d^{×}a dn dk,

with the outer three integrals over compact domains.

Proposition 2.2. If 𝔬 is elliptic, then

x ↦ K_{𝔬}(x, x) = ∑_{γ∈𝔬} f(x^{−1}γx)

is absolutely integrable over Z(𝔸)G(F)\G(𝔸), and its integral is

J_{𝔬}(f) = m(Z(𝔸)G(F)\G(𝔸)) ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γx) dx.

Proof. It suffices to prove the integrability assertion, since the ensuing formula for J_{𝔬}(f) follows just as in the case of compact quotient.

First note that for x lying in any fixed set Ω_{1}, which is compact modulo Z(𝔸), the sum defining K_{𝔬}(x, x) is actually finite and hence K_{𝔬} defines a nice, smooth function on Z(𝔸)G(F)\G(𝔸). Indeed, f(x^{−1}γx) ≠ 0 implies x^{−1}γx ∈ Ω_{f} = support of f. Thus γ itself must lie in a compact set modulo Z(𝔸), namely Ω_{1}Ω_{f}Ω_{1}^{−1}, and such a set can contain only finitely many γ in Z(F)\G(F).

On the other hand, if γ is elliptic, then f(x^{−1}γx) ≠ 0 implies (by Lemma 2.1) that αH(g) < d_{Ω_{f}}. I.e., the support of

K_{ell}(x, x) = ∑_{γ elliptic} f(x^{−1}γx)

LECTURE II. ARTHUR’S MODIFIED KERNELS I: THE GEOMETRIC TERMS 11

lies in the “doubly truncated” Siegel set

𝔖_{c}^{d} = Z(𝔸)C_{1}{h_{t}}C_{2}K with c < 2t < d_{Ω_{f}} .

Equivalently, the support of K_{ell} is compact modulo Z(𝔸), and the integrability is clear.

Remark. The argument above actually proves that the full elliptic contribution is absolutely integrable, i.e. , that

∫ |K_{ell}(x, x)| dx < ∞.

Thus, by the Lebesgue dominated convergence theorem, it follows that

∑_{𝔬 elliptic} ∫ |K_{𝔬}(x, x)| dx < ∞,

and hence that the sum

∑_{𝔬 elliptic} J_{𝔬}(f)

converges absolutely.

It remains now to prove Lemma 2.1. It might be tempting to do so for GL(2) by simply trying to multiply out some 2 × 2 matrices (as we did in [Ge, p 201]). However, this more quickly than not leads to an incomplete proof (as happened in fact in [Ge]), whereas the approach of [A1, p 362] works simply and generally, as we shall show below.

Proof of Lemma 2.1. Let

ρ : G → Gℓ(V)

denote the adjoint representation of G on the space of trace zero 2 × 2 matrices over F; its highest weight vector is Λ = α, the unique simple root of G with respect to the torus M. Fix a basis {e_{0}, e_{1}, e_{2}} of V such that

ρ(a)e_{j} = a^{Λ−jα}e_{j} for a ∈ M (j = 0, 1, 2)
ρ(n)e_{0} = e_{0}  for n ∈ N and
ρ(w)e_{0} = e_{2}

For ξ in V (𝔸), define

∥ ξ ∥ = ∏_{ν} ∥ ξ_{ν} ∥_{ν},

where, for finite ν, ∥ ξ_{ν} ∥_{ν} = max_{j}{|ξ_{ν}^{j}|_{ν}} (where (ξ_{ν}^{j}) denote the coordinates of ξ_{ν} with respect to the basis {e_{j}}), and for infinite ν, ∥ ξ_{ν} ∥_{ν} is defined by the Hilbert space structure that makes {e_{j}} into an orthonormal basis. (To make sense out of the infinite product, we restrict ξ to be “primitive”, i.e. , ∥ ξ_{ν} ∥_{ν} = 1 for all but finitely many ν.)

To prove the Lemma, suppose that

g^{−1}γng ∈ Ω (∗)

with n ∈ N(𝔸) and Ω compact modulo Z(𝔸). Write g = n_{1}ak, so that (∗) implies

a^{−1}n_{1}^{−1}γnn_{1}a ∈ KΩK.

Because the map g ↦ ρ(g)e_{0} is continuous, and ρ is trivial on Z(𝔸), it follows that

∥ ρ(a^{−1}n_{1}^{−1}γnn_{1}a) · e_{0} ∥ ≤ e^{2d_{0}}

for some d_{0} > 0. So suppose that γ does not belong to B(F). Then by Bruhat’s decomposition,

γ = b_{0}wn_{0}

for some b_{0} in B(F) and n_{0} in N(F). But for any b ∈ B(𝔸) and g ∈ G(𝔸), our assumptions on ρ and e_{0} imply

∥ ρ(gb) · e_{0} ∥ = e^{αH(b)} ∥ ρ(g) · e_{0} ∥.

Thus (with γ = b_{0}wn_{0}) we compute that

∥ ρ((a^{−1}n_{1}^{−1}b_{0}w)(n_{0}nn_{1}a)) · e_{0} ∥ = e^{αH(a)} ∥ ρ(a^{−1}n_{1}^{−1}b_{0}) · e_{2} ∥
= e^{2αH(a)} ∥ ρ(n^{∗}) · e_{2} ∥
≥ e^{2αH(a)} ∥ e_{2} ∥ = e^{2αH(g)}

(since H(g) = H(a), ρ(a) e_{2} = a^{−α} e_{2}, and ∥ ρ(n) e_{2} ∥ ≥ ∥ e_{2} ∥ for any n in N(𝔸)). Thus the Lemma is proved for d_{Ω} > d_{0}.

3. The Definition of KT_{𝔬}(x, y). For 𝔬 the (unique non-trivial) unipotent class in Z(F)\G(F), we saw in Lecture I that K_{𝔬}(x, y) is not integrable over the diagonal. A somewhat less trivial computation (see Lecture III) shows that the hyperbolic K_{𝔬}(x, y) are also not integrable. So let us finally introduce the required modifications of these kernels.

For P = B, let τ̂_{B} denote the characteristic function of the positive Weyl chamber

𝔞_{B}^{+} = { (r_{1}, r_{2}) ∈ 𝔞_{B} = 𝔞_{M} | r_{1} − r_{2} > 0 },

i.e. , the cone in ℝ^{2} where α(r_{1}, r_{2}) = r_{1} − r_{2} is positive. Then define, for any 𝔬 ∈ 𝔒 and T = (T_{1}, T_{2}) in 𝔞_{B}^{+},

(3.1)  kT_{𝔬} (x, f) = K_{𝔬}(x, x) − ∑_{δ∈B(F)\G(F)} K_{B,𝔬}(δx, δx) τ̂_{B}(H(δx) − T)

where

K_{B,𝔬}(x, y) = ∑_{γ∈𝔬∩Z(F)\M(F)} ∫_{N(𝔸)} f(x^{−1}γny) dn.

This is Arthur’s modified kernel kT_{𝔬} (x, f).

Note that if 𝔬 is elliptic, then 𝔬 ∩ M(F) = ∅. In this case K_{B,𝔬} ≡ 0, and

kT_{𝔬} (x, f) = K_{𝔬}(x, x).

In general, the term subtracted in 3.1 from K_{𝔬}(x, x) is the “correction term at ∞” which ends up making kT_{𝔬} (x, f) integrable. Before proving this is so, let us note the inductive nature of the definition of kT_{𝔬} (thereby making more plausible the appearance of K_{B,𝔬}). Set

(3.2)  K_{B}(x, y) = ∑_{𝔬} K_{B,𝔬}(x, y)

≡ ∑_{γ∈Z(F)\M(F)} ∫_{N(𝔸)} f(x^{−1}γny) dn.

Then K_{B}(x, y) is just the kernel of R(f) acting in L^{2}(Z(𝔸)N(𝔸)B(F)\G(𝔸)) in place of L^{2}(Z(𝔸)G(F)\G(𝔸). Indeed for φ in L^{2}(Z(𝔸)N(𝔸)B(F)\G(𝔸)),

R(f)φ(x) = ∫_{Z(𝔸)\G(𝔸)} f(x^{−1}y)φ(y) dy

= ∫_{Z(𝔸)N(𝔸)B(F)\G(𝔸)}( ∑_{Z(F)\M(F)} ∫_{N(𝔸)} f(x^{−1}γny) dn)φ(y) dy, (3.3)

as claimed. In general, for an arbitrary G, the definition of kT_{𝔬} (x, f) takes into account all the proper parabolics P of G, subtracting off from K_{𝔬}(x, x) not just one term (for the minimal parabolic B), but the sum

∑_{P ⊂G}(−1)^{dim A_{P} /Z} ∑_{δ∈P (F)\G(F)} K_{P,𝔬}(δx, δx) τ̂_{P} (H(δx) − T), (3.4)

where

K_{P} (x, y) = ∑_{𝔬∈𝔒} K_{P,𝔬}(x, y)

is the kernel of R_{P} (f) in L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸)). (Here A_{P} is the maximal split torus in M_{P}, the Levi component of P = M_{P} N_{P}, and R_{P} is the right regular representation of G(𝔸) in L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸)).)

Remark. Observe that

kT_{𝔬} (x, f) = K_{𝔬}(x, x).

for x in a compact (modulo Z(𝔸)) set Ω (how large depends on T). Indeed, for x in an appropriate such set Ω, τ̂_{B}(H(δx) − T) is identically zero^{∗}. Thus x is indeed a modification of K_{𝔬}(x, x) only “near infinity” (and in higher rank, more than one parabolic is needed to effect these modifications near infinity …).

∗Proof: Fix c and T such that α(T) > c. Fact 1 of Section 2 implies that in 𝔖_{c}, τ̂_{B}(H(δx)−T) ≠ 0 only for finitely many classes δ_{i} in B(F)\G(F). Then for each such class, τ̂_{B}(H(δ_{i}x) − T) = 0 on the compact set Ω_{i}^{T} in 𝔖_{c} where c < H(δ_{i}x) < α(T). Thus τ̂_{B}(H(δx) − T) is identically zero on Ω_{T} = ∩_{i=1}^{n}Ω_{i}^{T}.

4. The Integrability of kT_{𝔬} (x, f). Henceforth, let us assume that 𝔬 is unipotent or hyperbolic. To prove that kT_{𝔬} (x, f) is integrable over Z(𝔸)G(F)\G(𝔸), it suffices to show that it is integrable over a Siegel domain 𝔖_{c}. So pick any T ∈ 𝔞_{B}^{+}, and write

𝔖_{c} = B_{T} ∪ B̃_{T} ,

where

B_{T} = { g ∈ 𝔖_{c} | αH(g) > α(T) > 0 },

and B̃_{T} is its complement in 𝔖_{c}. Clearly kT_{𝔬} (x, f) is integrable over B̃_{T}, since

c < αH(g) ≤ α(T).

Thus it suffices to prove that kT_{𝔬} (x, f) is integrable over B_{T}. (N.B. In general, the partition of a Siegel domain requires more non-trivial “reduction theory” than is evident in the case of GL(2); this is what leads to the “geometric” combinatorial lemmas in §6 of [A2].)

Lemma 4.1. There exists a constant c_{0} such that αH(wn) ≤ c_{0} for all n ∈ N(𝔸); moreover, if γ in G(F) satisfies the inequality

αH(γx) > c_{0}

for some x in G(𝔸) with αH(x) > c_{0}, it must follow that γ ∈ B(F).

As the proof is similar to that of the crucial Lemma 2.1 in §2 of this Lecture, we leave the details to the reader (see [A1, pp 334–335]).

Now fix T in 𝔞_{B}^{+} such that α(T) > c_{0}. Then for x in B_{T}, Lemma 4.1 implies that

τ̂_{B}(H(δx) − T) ≠ 0

(if and) only if δ ∈ B(F). Thus (for such x),

∑_{γ∈𝔬} f(x^{−1}γx) = ∑_{δ∈B(F)\G(F)} ∑_{γ∈𝔬} f(x^{−1}δ^{−1}γδx)τ̂_{B}(H(δx) − T),

and hence

kT_{𝔬} (x, f) = ∑_{δ∈B(F)\G(F)}{∑_{γ∈𝔬} f(x^{−1}δ^{−1}γδx)

− ∫_{N(𝔸)} ∑_{μ∈M(F)∩𝔬} f(x^{−1}δ^{−1}μnδx) dn}τ̂_{B}(H(δx) − T)

Next fix T such that α(T) > max{c_{0}, d_{Ω_{f}} }, with d_{Ω_{f}} as in Lemma 2.1. Then by Lemma 2.1, we may replace the sum here over γ ∈ 𝔬 by a sum over

B(F) ∩ 𝔬 = N(F) · M(F) ∩ 𝔬

(This last equality holds since the class of γ in B(F) is determined by its semi-simple part, i.e. its M(F)-part.) Thus we get

kT_{𝔬} (x, f) = ∑_{δ∈B(F)\G(F)} ∑_{μ∈M(F)∩𝔬}{ ∑_{ν∈N(F)} f(x^{−1}δ^{−1}μνδx)

− ∫_{N(𝔸)} f(x^{−1}δ^{−1}μnδx) dn}τ̂_{B}(H(δx) − T) (4.1)

= ∑_{δ} ∑_{μ} ∑_{ν∈N(F)−{1}} F̂_{δx,μ}(ν) τ̂_{B}(H(δx) − T)

where

F_{x,μ}(n) = f(x^{−1}μnx)

defines a Schwartz-Bruhat function on N(𝔸) ≈ 𝔸 with Fourier transform

F̂_{x,μ}(ν) = ∫_{N(𝔸)} f(x^{−1}μnx)ψ_{N}(nν) dn

(Here ψ_{N}(({1 \atop 0} {x \atop 1})) = ψ(x) for some fixed non-trivial character on F \𝔸, and we have applied the Poisson summation formula to F_{δx,μ}.) Finally we are ready to prove

Proposition 4.2. For any 𝔬 ∈ 𝔒, kT_{𝔬} (x, f) is absolutely integrable over

Z(𝔸)G(F)\G(𝔸).

Proof. We may assume 𝔬 is not elliptic. By the above arguments, it suffices to check the integrability over B_{T}, with α(T) > max{c_{0}, d_{Ω_{f}} } as above. As in that domain formula (4.1) holds, we conclude

(4.2)  ∫ |kT_{𝔬} (x, f)| dx ≤
∫Z(𝔸)B(F)\G(𝔸) ∑_{μ∈M(F)∩𝔬} ∑_{ν∈N(F)−{1}} |F̂_{x,μ}(ν)| τ̂_{B}(H(x) − T) dx.

Note that for any μ = ({α \atop 0} {0 \atop 1}) in M(F) ∩ 𝔬, ν = ({1 \atop 0} {w \atop 1} ) in N(F) − \{1\}, and x = znh_{t}mk,

F̂_{x,μ}(ν) = ∫_{𝔸} f (x^{−1}({α \atop 0} {0 \atop 1})({1 \atop 0} {y \atop 1} )x) ψ(yw) dy
= e^{2t}F̂_{k^{∗},μ}(e^{2t}ν)

with k^{∗} = (h_{t}^{−1}nh_{t})mk. (Here e^{2t}ν = ({1 \atop 0} {e^{2t}w \atop 1} ), and we confuse N(F) with F.) Note also that the variable x in (4.2) runs through a Siegel set. Thus k^{∗} runs through a compact set, and we have, for any N > 0

∑_{ν≠1} F̂_{k^{∗},μ}(e^{2t}ν) ≤ c_{N}(e^{−2t})^{N}

Applying Iwasawa’s decomposition (2.3) to (4.2), we then conclude that

∫ |kT_{𝔬} (x, f)| dx ≤ C ∫_{N(F)\N(𝔸)} ∫_{F^{×}\𝔸^{1}} ∫_{K}(∫_{α(T)}^{∞} e^{−2tN} dt) dk d^{×}a dn < ∞

as required.

Summing up, we have

Theorem 4.3. For α(T) sufficiently large,

∑_{𝔬∈𝔒} ∫_{Z(𝔸)G(F)\G(𝔸)} |kT_{𝔬} (x, f)| dx < ∞.

Hence the left side ∑_{𝔬∈𝔒} J T_{𝔬} (f) of (1) of Lecture I is defined and absolutely convergent.

Indeed, it suffices to remark that for a given f ∈ C_{c}^{∞}(Z(𝔸)\G(𝔸)), kT_{𝔬} (x, f) is identically zero except for finitely many hyperbolic 𝔬. This is clear from the defining equation (3.1), and the fact that x^{−1}δ^{−1}({α \atop 0} {0 \atop 1})δx and x^{−1}δ^{−1}({α \atop 0} {0 \atop 1})nδx can belong to the fixed compact mod Z(𝔸) set Ω_{f} for only finitely many α; to see this, observe that conjugating by δx preserves eigenvalues. Thus the sum over 𝔬 in 𝔒 in Theorem 4.3 is infinite only over the elliptic classes, where we already know the relevant sum ∑ ∫ |K_{𝔬}(x, f)| dx is finite (see the Remark after the proof of Proposition 2.2).

Concluding Remarks. (a) The terms J T_{𝔬} (f) are indeed generalizations of the orbital integrals appearing on the geometric side of Selberg’s compact quotient trace formula. Indeed, for 𝔬 elliptic (which is automatic for G anisotropic), J T_{𝔬} (f) reduces to a multiple of the usual orbital integral ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γx) dx.

We have seen—for GL(2)—that the kernel K_{𝔬}(x, x) is already integrable on 𝔖_{c} if 𝔬 is an elliptic class (as opposed to hyperbolic or unipotent). For general G, the proper distinction is between those 𝔬 which never intersect any proper parabolic subgroup subgroup P of G and those that do. Indeed Arthur does not talk about “elliptic” 𝔬 at all, but rather defines a modified kernel

kT_{𝔬} (x, f)

for any 𝔬, and proves that kT_{𝔬} (x, f) is integrable. But of course—as we have already observed—kT_{𝔬} (x, f) = K_{𝔬}(x, x) precisely when 𝔬 is elliptic in this sense (that it intersects no proper P …).

References

[A1] Arthur, J., The Selberg trace formula for groups of F-rank one, Annals of Math. 100 (1974), pp. 236–385.

[A2] Arthur, J., A trace formula for reductive groups II: Terms associated to classes in G(ℚ), Duke Math. J., 45, No. 4 (1978), pp. 911–952.

[A7] Arthur, J., The trace formula for reductive groups, Lectures for Journées Automorphes, Dijon (1981).

[Ge] Gelbart, S., Automorphic Forms on Adele Groups, Annals of Math. Studies No. 83, Princeton University Press, Princeton NJ, 1975.

[GJ] Gelbart, S. and Jacquet, H., Forms of GL(2) from the analytic point of view, in Proc. Symp. Pure Math. (Corvallis) , vol. 33 Part I, (1979), pp 213–251.

[Go] Godement, R., Domaines fondamentaux des groupes arithmetiques, Seminaire Bourbaki 257, W. A. Benjamin, Inc., New York, (1963).