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

In order to describe the right hand side of Arthur’s trace formula, we need first to recall the spectral expansion of the kernel

K_{f}(x, y) = ∑_{χ∈𝔛} K_{χ}(x, y).

1. The Spectral Kernels K_{χ}. Henceforth, by a “Levi subgroup M of G” we understand the Levi component of a parabolic subgroup P of G; for any such M, we have

M^{1}(𝔸) = {m ∈ M(𝔸): |λ(m)|_{𝔸} = 1 ∀λ ∈ X(M)_{F} }.

In general, the decomposition of

L^{2}(Z(𝔸)G(F)\G(𝔸))

into right G(𝔸)-invariant subspaces is determined by spectral data χ = {(M, r)} ∈ 𝔛, where the pair (M, r) consists of a Levi subgroup M of G and a cuspidal representation r of Z(𝔸)\M(𝔸); the class {(M, r)} derives from the equivalence relation

(M, r) ∼ (M′, r)

if and only if M is conjugate to M′ by a Weyl group element w, and r′ = r^{w} on Z(𝔸)\M^{1}(𝔸). Then to each such cuspidal datum χ = {(M, r)} is associated a subspace L_{χ}^{2}, such that

L^{2}(Z(𝔸)G(F)\G(𝔸)) = ⨁_{χ} L_{χ}^{2},

and the corresponding decomposition of kernels

K_{f}(x, y) = ∑_{χ} K_{χ}(x, y)

is explicitly describable in terms of Eisenstein series.

For G = GL(2), cuspidal data comes in two possible forms:

Typeset by AMS-TEX

Case (i): M = G, r = a cuspidal representation π of Z(𝔸)\G(𝔸). In this case, L_{χ}^{2} = L_{π}, the irreducible subrepresentation of L_{0}^{2}(Z(𝔸)G(F)\G(𝔸)) realizing π, and

(1.1)  K_{χ}(x, y) = K_{π}(x, y) = ∑_{\{ϕ\}= o.n. basis of L_{π}}R(f)ϕ(x)\overline{ϕ(y)}

Case (ii): M = \{( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} )\}, r ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ) = μ (\frac{a_{1}}{a_{2}} ) with μ a character of F^{x}\𝔸^{1}

(identified with the pair (M, μ^{−1} ). In this case,

(1.2) K_{χ}(x, y) = K_{μ}(x, y) = ∑_{{ϕ_{μ}}} ∫_{−∞}^{∞} E(x, ρ(μ, it)(f)ϕ_{μ}, it, μ) E(y, ϕ_{μ}, it, μ) dt.

where {ϕ_{μ}} runs through an ON basis for the induced representation space

Ind_{B(𝔸)}^{G(𝔸)} μ (\frac{a_{1}}{a_{2}} ) = ρ(μ, o),

consisting of functions ϕ : G(𝔸) ⟶ ℂ such that

ϕ (( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) g) = μ (\frac{a_{1}}{a_{2}} ) |\frac{a_{1}}{a_{2}} |_{𝔸}^{1/2}ϕ(g)

for all ( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) in B(𝔸), and ∫_{K} |ϕ(k)|^{2} dk < ∞; ρ(μ, s) denotes the induced

representation space Ind_{B(𝔸)}^{G(𝔸)} μ (\frac{a_{1}}{a_{2}} ) |\frac{a_{1}}{a_{2}} |_{𝔸}^{s}, and E(g, ϕ_{μ}, s, μ) is the Eisenstein series

∑_{γ∈B(F)\G(F)} ϕ_{μ}^{s}(γg)

attached to the function

ϕ_{μ}^{s}(g) = ϕ_{μ}(g)e^{sα(H(g))}

in ρ(μ, s).

Remarks. (i) The series defining E(g, ϕ_{μ}, s, μ) converges only for Re(s) > 1/2, but E(g) itself has a meromorphic continuation to all of ℂ, with a possible pole at s = 1/2. These and other facts from the theory of Eisenstein series we shall simply assume, referring the reader to [GJ] for more details.

(ii) If μ^{2} ≡ 1, then (and only then) K_{χ}(x, y) has an additional “discrete” term, namely

(1.3)  (R(f)μ)(x)\overline{μ(y)} = ∫_{Z(𝔸)\G(𝔸)} f(g)μ(det g) dg μ(det x)\overline{μ(det y)}.

(this comes from the residue of the corresponding Eisenstein series at s = 1/2 …).

(iii) Above, K_{χ}(x, y) represents the kernel of R(f) acting in the space L_{χ}^{2}, which (for any χ = {(M, μ)}) consists of all φ(g) which are orthogonal to the space of cusp forms L_{0}^{2}(Z(𝔸)G(F)\G(𝔸)), and such that for almost every x in G(𝔸), the projection of the function

φ_{N}^{x}(m) = ∫_{N(F)\N(𝔸)} φ(nmx) dn,

onto the space L^{2}(Z(𝔸)M(F)\M^{1}(𝔸)) transforms under Z(𝔸)\M^{1}(𝔸) as a sum or representations r, where (M, r) is a pair in χ.

The beauty of this description is that it generalizes to a general reductive group G, as we shall now describe.

For any fixed parabolic P = M_{P} N_{P} in a reductive G, one may describe a decomposition

(1.4)  L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸)) = L^{2}(P) = ⨁_{χ∈𝔛} L^{2}(P)_{χ},

where L^{2}(P)_{χ} consists of those φ in L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸)) with the following property: for each standard parabolic subgroup B of G, with B ⊂ P, and almost every x in G(𝔸), the projection of the function

m ⟶ φ_{B,x}(m) = ∫_{N_{B}(F)\N_{B}(𝔸)} φ(nmx) dn

onto the space of cusp forms in L^{2}(Z(𝔸)M_{B}(F)\M_{B}^{1}(𝔸)) transforms under M_{B}^{1}(𝔸) as a sum of representations r_{B}, in which (M_{B}, r_{B}) is a pair in χ. If there is no such pair in χ, φ_{B,x} will be orthogonal to the space of cusp forms on Z(𝔸)M_{B}(F)\M_{B}^{1}(𝔸) (as happens above for G = GL(2), P = G = B, and χ = {(M, μ)}).

Remark. For any fixed χ in 𝔛, let P_{χ} be the set of standard parabolics B, where (M_{B}, r_{B}) belongs to χ. Then from the theory of Eisenstein series it follows

L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸))_{χ}

will be zero unless there is a group in P_{χ} contained in P.

Let us return now to the decomposition

(1.4)  L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸)) = L^{2}(P) = ⨁_{χ∈𝔛} L^{2}(P)_{χ},

described above. If we let K_{P,χ}(x, y) denote the integral kernel of the restriction of R_{P} (f) to L^{2}(P)_{χ}, then one can still—in general—write down a formula for K_{P,χ}(x, y) in terms of Eisenstein series. Clearly

(1.5)  ∑_{𝔬∈𝔒} K_{P,𝔬}(x, y) = ∑_{χ∈𝔛} K_{P,χ}(x, y),

as each side equals the integral kernel of R_{P} (f) (see §3 of Lecture II, where we computed the “geometric” expression for the kernel K_{P}, for G = GL(2) and P = Borel).

Summing Up. We have

K_{f}(x, y) = ∑_{χ∈x} K_{χ}(x, y)

and (for any parabolic P of G),

K_{P} (χ, y) = ∑_{χ∈x} K_{P,χ}(x, y).

Concluding Remarks. (a) When P = G,

K_{P} (x, y) = K_{G}(x, y) = K(x, y)

and

K_{P,χ}(x, y) = K_{χ}(x, y).

Thus the first spectral expansion is a special case of the second.

(b) For G = GL(2), and P = B (the Borel), we have L^{2}(P)_{χ} = 0 if χ = (G, π). On the other hand, if χ = {(M_{P} , μ)}, then L^{2}(P)_{χ} just consists of φ in L^{2}(Z(𝔸)N_{P} (𝔸)M_{P} (F)\G(𝔸), which under left action by M^{1}(𝔸) transforms according to a sum of μ and μ^{−1}. The corresponding kernel K_{B,χ}(x, y) is given by the formula

(1.6)  K_{B,χ}(x, y) = ∫_{Z(𝔸)M(F)\M^{1}(𝔸)} K_{B}(x, my)(μ(m) + μ^{w}(m)) dm.

This is easily checked by computing the composition of R(f) with the projection operator P_{χ} defined on L^{2}(Z(𝔸)N(𝔸)M(F)\G(𝔸)) by

(P_{χ}φ)(y) = ∫_{Z(𝔸)M(F)\M^{1}(𝔸)} φ(my)(μ(m) + μ^{w}(m)) dm.

Moreover, an analogue of (1.6) (without the subscript B) holds for the kernel K_{χ}(x, y).

(c) The series

∑ K_{χ}(x, y) = ∑ K_{G,χ}(x, y)

and

∑ K_{P,χ}(x, y)

converge absolutely (to K(x, y) and K_{P} (x, y) respectively).

(d) In general, for any G and χ = {(M, r)}

(1.7) ∑ K_{G,χ}(x, y) =

∑_{P} \frac{1}{n(A_{P})} ∑_{σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} ∑_{ϕ} E(χ, ρ(σ, λ)(f)ϕ, λ)\overline{E(y, ϕ, λ)} dλ,

where: the sum over P is over “associated” parabolics in G, n(A_{P}) is the number of chambers in 𝔞_{P}, 𝔞_{P}^{∗} = X(M_{P}) ⊗ ℝ is dual to 𝔞_{P} = Hom(X(M_{P})_{F} , ℝ), ρ(σ, λ) is the right regular representation in Ind_{P (𝔸)}^{G(𝔸)} σ ⊗ e^{λ(H_{P} (p))}, ϕ runs through a suitable K-finite) basis for ρ(σ, λ), and—critically, σ runs through (classes of) irreducible unitary representations of M(𝔸) such that functions in Ind_{P (𝔸)}^{G(𝔸)} σ belong to L^{2}(P)_{χ}. Note that for χ = {(G, π)}, (1.3) just reduces to

∑_{ϕ} R(f)ϕ(x)\overline{ϕ(y)},

since L^{2}(P)_{χ} ≡ 0 unless P = G. Thus our original formula for K_{π}(x, y) is indeed consistent with the statement that any K_{P,χ}(x, y) is expressible in terms of Eisenstein series (but now the Eisenstein series in question belong to ϕ in L_{π}, viewed as induced from themselves on P = G). At the same time, formula (1.7) is also general enough to include the extra term R(f)(μ)(x)\overline{μ(y)} in case G = GL(2), and χ = {(M, μ)} with μ = μ^{−1}; indeed, the parabolic P = B in (1.7) contributes the leading term (1.2) to K_{χ}(x, y), while the parabolic P = G contributes the crucial additional term (1.3).

2. Modified Kernels and Truncated Kernels. Recall the identity

(1.5)  ∑_{𝔬∈𝔒} K_{P,𝔬}(x, y) = ∑_{χ∈𝔛} K_{P,χ}(x, y),

introduced towards the end of the last section. By complete analogy with the geometric side (compare formulas (3.1) and (3.4) of Lecture II), Arthur defines the modified spectral kernel functions

(2.1) k_{χ}^{T} (x, f) = ∑_{P ⊂G}(−1)^{dim(A_{P} /Z)} ∑_{δ∈P (F)\G(F)} K_{P,χ}(δx, δx)τ̂_{P} (H_{P} (δx) − T),

which in the case of GL(2) look like

k_{χ}^{T} (x, f) = K_{χ}(x, x) − ∑_{δ∈B(F)\G(F)} K_{B,χ}(δx, δx)τ̂_{B}(H_{B}(δx) − T).

Because of the identity (1.5), one clearly has

(2.2)  ∑_{𝔬∈𝔒} k_{𝔬}^{T} (x, f) = ∑_{χ∈𝔛} k_{χ}^{T} (x, f),

i.e., the modification of the geometric expression for the kernel of R(f) is equal to the modification of the spectral expression for this kernel.

Despite its simplicity, this identity (2.2) is the starting point of Arthur’s trace formula in the case of non-compact quotient. Of course, we emphasize (again!) that neither side of (2.2) represents (in general) the kernel of R(f). However, we have already seen in Lecture II that each of the functions k_{𝔬}^{T} (x, f) is (absolutely) integrable over Z(𝔸)G(F)\G(𝔸), and that the sum of the resulting distributions

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

also converges absolutely. Thus by (2.2), we have shown that

(2.3)  ∑_{𝔬} J_{𝔬}^{T} (f) = ∫_{Z(𝔸)G(F)\G(𝔸)} ∑_{χ} k_{χ}^{T} (x, f) dx,

i.e., to obtain the first form of Arthur’s trace formula, we need “only” show that the integral (on the right side of (2.3) may be taken inside the sum over χ; in particular, each k_{χ}^{T} (x, f) is integrable, and the sum of the resulting distributions

(2.4)  J_{χ}^{T} (f) = ∫ k_{χ}^{T} (x, f) dx

converges absolutely to the left side of (2.3).

To prove the absolute integrability of each k_{χ}^{T} (x, f), and of the sum of such kernels, it turns out to be necessary to introduce a truncation operator on G(F)\G(𝔸). Given T in 𝔞^{+} as before, the truncation of a continuous function φ(x) on Z(𝔸)G(F)\G(𝔸) is the function

(2.5)  Λ^{T} φ(x) = φ(x) − ∑_{δ∈B(F)\G(F)} φ_{N}(δx)τ̂_{B}(H(δx) − T),

where

φ_{N}(δx) = ∫_{N(F)\N(𝔸)} φ(nx) dn

denotes the constant term of φ. (Again, for x ∈ 𝔖_{c}, the sum over δ is finite, by Fact 1 of Lecture II …) In general, for arbitrary reductive G,

(2.6)  Λ^{T} φ(x) = ∑_{P ⊂G}(−1)^{dim(A_{P} /Z)} ∑_{δ∈P (F)\G(F)} φ_{N_{P}} (δx)τ̂_{P} (H(δx) − T).

Note that if φ(x) is cuspidal, i.e., φ_{N_{P}} = 0 for every N_{P}, then

Λ^{T} φ = φ.

Of course, for a general automorphic φ, this is no longer true; however, the whole idea of truncation is that Λ^{T} φ still equals φ on a large part of 𝔖_{c} (how large depends on T), and suitably modifies φ “near infinity” so as to be integrable over all of 𝔖_{c}. In particular, for given “large” T, there is a compact (mod Z(𝔸)) set Ω such that

Λ^{T} φ ≡ φ on Ω.

(Both this statement—and its proof—resembles its analogue for the modified kernels k_{𝔬}^{T} (x, f); see Remark 3.5 of Lecture II.)

Now for any χ in 𝔛, let Λ_{2}^{T} K_{χ}(x, x) denote the function obtained by truncating the function

K_{G,χ}(x, y) = K_{χ}(x, y)

with respect to the second variable and then setting y = x. The strategy Arthur follows in [A3] to prove that

(2.7)  ∫ (∑_{χ} k_{χ}^{T} (x, f)) dx = ∑_{χ} ∫ k_{χ}^{T} (x, f) dx

is the following: Using properties of Λ^{T}, he first shows that

(2.8)  ∫ ∑_{χ} Λ_{2}^{T} k_{χ}(x, x) dx = ∑_{χ} ∫ Λ_{2}^{T} k_{χ}(x, x) dx

with

∑_{χ} ∫ |Λ_{2}^{T} k_{χ}(x, x)| dx < ∞

and

∫ |∑_{χ} Λ_{2}^{T} k_{χ}(x, x)| dx < ∞;

then he shows that for sufficiently large T,

(2.9)  ∫ ∑_{χ} |Λ_{2}^{T} k_{χ}(x, x) − k_{χ}^{T} (x, f)| dx = 0.

From these facts it follows that ∑_{𝔬} J_{𝔬}^{T} (f) converges absolutely to ∑_{χ} J_{χ}^{T} (f), where for sufficiently large T, J_{χ}^{T} (f) is given by either (absolutely convergent) integral

∫ k_{χ}^{T} (x, f) dx or ∫ Λ_{2}^{T} K_{χ}(x, x) dx.

In particular, the first form of Arthur’s trace formula is established.

Concluding Remarks. Set

K_{cusp}(x, y) = ∑_{χ∈𝔛(G)} K_{χ}(x, y)

where 𝔛(G) denotes the set of cuspidal data {(M, σ)} with M = G, and R_{0} be the regular representation of G(𝔸) restricted to

L_{0}^{2} = ⨁_{χ∈𝔛(G)} L_{χ}^{2}.

Then for any T,

(2.10)  Λ_{2}^{T} K_{cusp}(x, y) = K_{cusp}(x, y),

this kernel is integrable over the diagonal, and equals to R_{0}(f). Thus Arthur’s first form of the trace formula gives the more familiar trace formula

(2.11)  tr R_{0}(f) = ∑_{𝔬∈𝔒} J_{𝔬}^{T} (f) − ∑_{𝔛\𝔛(G)} J_{χ}^{T} (f).

Here only the proof of (2.10) is elementary; it results from the observation that y ⟶ K_{cusp}(x, y) is cuspidal (see the beginning of §4 below). The remaining facts are discussed in [A1], at least for the case of rank 1. To prove that tr R_{0}(f) exists and equals K_{cusp}(x, x) dx one uses arguments similar to those used in the (earlier) proof that ∫ k_{𝔬}^{T} (x, f) dx < ∞.

3. Proof of (2.8) and the Absolute Convergence of ∑_{χ} ∫ Λ_{2}^{T} K_{χ}(x, y) dx <

∞. When Arthur described the spectral expansion

(3.1)  K(x, y) = ∑ K_{χ}(x, y).

in [A2], he stressed its absolute convergence by way of the following:

Lemma 3.2. (see Lemma 4.4 of [A2]) There exists an N such that for any differential operators D_{1} and D_{2} in 𝔘(𝔤),

∑_{χ} ∑_{P} n(A)^{−1} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} |∑_{ϕ_{χ}} D_{1}E(x, ρ(σ, λ)(f)(ϕ_{χ}), λ)\overline{D_{2}E(y, ϕ_{χ}, λ)}| dλ

≤ c(D_{1}, D_{2})∥x∥^{N}∥y∥^{N}.

Here ∥x∥ denotes a “norm” or “height” function on G(𝔸), which we may (for GL(2)) take to be ∏_{v} ∥x_{v}∥, with ∥x_{v}∥ = sup{|(x_{v})_{ij}|_{v}, |(x_{v}^{−1})_{ij}|_{v}}. This norm satisfies the properties ∥x∥ = ∥x^{−1}∥, ∥x_{1}x_{2}∥ ≤ ∥x_{1}∥∥x_{2}∥, and ∥x > ∥c_{1}, for some c_{1}; moreover, it is commensurable with e^{α(H(x))}, at least for x in a fixed Siegel domain.

The above Lemma is essentially part of Langlands’ theory of Eisenstein series; from it, we get

(3.3)  |D_{1,x}D_{2,y}K_{χ}(x, y)| ≤ c(D_{1}, D_{2})∥x∥^{N}∥y∥^{N},

with a similar identity for K(x, y) itself . (Here D_{2,y} indicates that D_{2} is operating only in the y variable, etc..) These identities say that K(x, y) and K_{χ}(x, y) are “slowly increasing” in each variable. It is by way of these facts that the finiteness of

∑_{χ} ∫ |Λ_{2}^{T} K(x, x)| dx

is established.

Lemma 3.4. (see Lemma 1.4 of [A3]) The truncation operator transforms “slowly increasing functions” φ into “rapidly decreasing” ones; more precisely, suppose φ in C^{∞}(Z(𝔸)G(F)\G(𝔸)) is such that for some N (the “degree of slow increase”),

|Dφ(x)| ≤ C_{D}∥x∥^{N} for all x ∈ 𝔖_{c}

and differential operators D. Then for any M

|Λ^{T} φ(x)| ≤ C_{M}∥x∥^{−M}

for x in 𝔖_{c} (and some C_{M}).

Remarks. (a) Since we are not proving Lemma (3.5), let us at least explain why it is plausible for G = GL(2). In this case, it is clear from Lemma 2 of Lecture II, that for x “near infinity”, in particular, such that α(H(x) > α(T) > C_{0},

Λ^{T} φ(x) = φ(x) − φ_{N}(x),

Thus Λ^{T} removes the constant term form ϕ, making it rapidly decreasing at ∞ (think roughly of a classical modular form for SL_{2}(ℤ) with vanishing constant term …).

(b) What Lemma 1.4 of [A3] actually asserts is that for any pair of positive integers M and N, we can choose D_{1}, … D_{n} such that for any x in 𝔖_{c},

(3.5)  |Λ^{T} φ(x)| ≤ ∑_{i} (sup_{x′} |D_{i}φ(x′)| · ∥x′∥^{−N}) · ∥x′∥^{−M}

But if φ happens to be slowly increasing of degree N, along with its derivatives, then the expression in parentheses is bounded, and so Lemma 3.4 immediately follows. It is this latter form of the Lemma—that is, (3.5)—that we shall now apply.

Proposition 3.5. The truncated kernels Λ_{2}^{T} K(x, x) and Λ_{2}^{T} K_{χ}(x, x) are absolutely integrable, and

∫ Λ_{2}^{T} K(x, x) dx = ∑_{χ} ∫ Λ_{2}^{T} K_{χ}(x, x) dx.

Proof. Let us show that for any M, there exists constants c′ and c_{χ} such that for x in 𝔖_{c},

Λ_{2}^{T} K(x, x) ≤ c′∥x∥^{N−M} and Λ_{2}^{T} K_{χ}(x, x) ≤ c′_{χ}∥x∥^{N−M}.

Since ∥x∥ ≍ e^{α(H(x))} on any Siegel domain, this will suffice, by Iwasawa’s decomposition, to prove that

∫ |Λ_{2}^{T} K(x, x)| dx < ∞,

and hence by (Lebesgue’s) dominated convergence theorem, that

∫ Λ_{2}^{T} K(x, x) dx = ∑_{χ} ∫ Λ_{2}^{T} K(x, x) dx,

an absolutely convergent sum.

Applying Lemma 3.4 (in the form of identity (3.5)) to the second variable of K(x, y) yields

|Λ_{2}^{T} K(x, y)| ≤ C_{0}∥y∥^{−M} · sup_{y′} sup_{D_{1},...D_{n}}(|D_{i,y}K(x, y′)|∥y′∥^{−N}),

which by (3.3) is dominated by

C_{0}∥y∥^{−M} sup_{y′,i}(C(1, D_{i})∥x∥^{N}∥y′∥^{N})∥y′∥^{−N} = C_{0}∥y∥^{−N} sup_{i}(C(1, D_{i}))∥x∥^{N}.

So now setting x = y gives the desired conclusions (for K(x, x)), and for K_{χ}(x, x) one argues similarly.

4. Proof of (2.9) Relating ∫ Λ_{2}^{T} K_{χ}(x, x) dx to ∫ k_{χ}^{T} (x, t) dx. Note first that when χ = (G, π), there is nothing to prove. Indeed, if K_{π} is the kernel of R(f) restricted to any cuspidal subspace L_{π} of L^{2}(Z(𝔸)G(F)\G(𝔸)), then

y ⟶ K_{π}(x, y)

is itself cuspidal: for any φ^{∗} in (L_{0}^{2})^{⊥},

∫ K_{π}(x, y)\overline{φ^{∗}(y)} dy = R(f) ∘ (Projection onto L_{π} ⊂ L_{0}^{2}) = 0.

Thus it follows Λ_{2}^{T} K_{π}(x, y) = K_{π}(x, y). On the other hand, as recalled in §1, K_{P,χ}(x, y) ≡ 0 for all proper P in G (when χ = (G, π)). Thus we also have (from the definition (2.1) of k_{χ}^{T} (x, f)) that k_{χ}^{T} (x, f) = K_{G,χ}(x, x) = K_{π}(x, x), i.e., Λ_{2}^{T} K_{π}(x, x) ≡ k_{χ}^{T} (x, f), and there is nothing to prove.

Henceforth, we shall fix G = GL(2), and assume χ = {(M, σ)} with M =

\{( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )\}, and σ ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) = μ(a/b) (or rather its class, relative to the relation

μ ∼ μ^{−1} in (F^{x}\𝔸^{1})^{∧}). Adapting the general arguments of §2 of [A3] to this case, we shall now (at least) prove that

∫ |Λ_{2}^{T} K_{χ}(x, x) − k_{χ}^{T} (x, f)| dx = 0

for sufficiently large T.

By definition,

− (Λ_{2}^{T} K_{χ}(x, x) − k_{χ}^{T} (x, f)) =

∑_{δ∈B(F)\G(F)} ∫_{N(F)\N(𝔸)} K_{χ}(x, nδx)τ̂_{B}(H_{B}(δx) − T) dn

− ∑_{δ∈B(F)\G(F)} K_{B,χ}(δx, δx)τ̂_{B}(H_{B}(δx) − T).

Note that K_{χ}(x, nδx) = K_{χ}(δx, nδx) (since E(x, ϕ) is left G(F)-invariant). Therefore the above difference equals

(4.1) ∑_{δ∈B(F)\G(F)} τ̂_{B}(H_{B}(δx) − T) {∫_{N(F)\N(𝔸)} K_{χ}(δx, nδx) dn − K_{B,χ}(δx, δx)} .

Next recall our expression (1.6) for K_{B,χ} (and K_{G,χ}) in the Concluding Remark (b) of §1. Applying it to (4.1), with y = δx, we get that the expression in parenthesis equals

∫_{N(F)\N(𝔸)} K_{χ}(y, ny) dn − K_{B,χ}(y, y) =

∫_{Z(𝔸)M(F)\M^{1}(𝔸)} {∫_{N(F)\N(𝔸)}(K(y, nmy) − K_{B}(y, my)) dn} ×
μ(m) + μ^{−1}(m) dm.

Applying the definition of K and K_{B}, we find, in turn, that the last expression in parentheses equals

∫_{N(F)\N(𝔸)} ∑_{γ∈Z(F)\G(F)}f(y^{−1}γ n m y) dn − ∫_{N(𝔸)} ∑_{μ∈Z(F)\M(F)}f(y^{−1}γ n m y) dn

= ∫_{N(𝔸)} (  ∑_{γ∈N(F)Z(F)\G(F)}f(y^{−1}γ n m y) − ∫_{N(𝔸)} ∑_{μ∈Z(F)\M(F)}f(y^{−1}γ n m y)) dn

Now apply Bruhat’s decomposition

Z(F)N(F)\G(F) = Z(F)\M(F) ∪ (Z(F)\M(F))wN(F)

to the first summation above. The result is that the entire last expression equals

∫_{N(𝔸)} ∑_{ν∈N(F)} ∑_{μ∈Z(F)\M(F)} f(y^{−1}μ w ν n m y) dn.

Using this, it remains to show that

∫ |Λ_{2}^{T} K_{χ}(x, x) − k_{χ}^{T} (x, f)| dx

≤ ∫_{Z(𝔸)G(F)\G(𝔸)} ∑_{δ∈B(F)\G(F)} τ̂_{B}(H(δx) − T) ×

∫_{Z(𝔸)M(F)\M^{1}(𝔸)} ∫_{N(𝔸)} ∑ ∑ f(x^{−1}δμ w ν n mδχ)(μ(m) + μ^{−1}(m) dm dχ.

= ∫_{Z(𝔸)B(F)\G(𝔸)} τ̂_{B}(H(y) − T) ×

(4.2)  (∫∫ ∑ ∑ f(y^{−1}δμ w ν n m y)(μ(m) + μ^{−1}(m)) dm dy

= 0 for sufficiently large T .

So suppose y is such that this expression (4.2) is not zero. Then our hypothesis on f implies

y^{−1}μ w ν n m y

belongs to the compact (mod Z(𝔸)) subset Ω_{f} of G(𝔸). Writing out the Iwasawa decomposition n_{1}ak for y implies

g = a^{−1}n_{1}^{−1}ν w μ n m n_{1}a ∈ Ω′_{f}

with Ω′_{f} the compact ( mod Z(𝔸)) set KΩ_{f}K, i.e., α(H(g)) remains bounded (from below and above).

On the other hand, write

g = a^{−1}n_{1}^{−1}ν w μ n m n_{1}a
= (n_{1}^{−1}ν)^{a^{−1}}a^{−1})w μ m a (n^{m^{−1}}n_{1})^{a^{−1}}
= n^{∗}a^{∗}wn′

where n^{∗}, n′ ∈ N(𝔸) and a^{∗} = a^{−1}a^{w}(μm)^{m}. Then

c < H(g) = H(a^{∗}) + H(wn′) = −2H(a) + H(wn′)

implies

−2H(a) > c − H(wn′),

i.e.

(4.3)  H(a) < \frac{H(wn′)}{2} − \frac{c}{2} < \frac{c_{0} − c}{2} = c′

(using Lemma 4.1 of Lecture II).

To complete the proof, take T “sufficiently large” to mean that α(T) > c′. Then the condition τ̂_{B}(H(y) − T) = τ̂_{B}(H(a) − T) ≠ 0 will be incompatible with (4.3), i.e. (4.2) will be identically zero, as claimed.

5. An Alternate Strategy?. In any treatment of the trace formula prior to the 1978 paper [A2] (see [A1]), no modified kernels appear. For example, in [GJ] one simply and (formally) truncates both sides of the identity

K(x, y) = ∑_{γ} f(x^{−1}γy) = ∑ K_{χ}(x, y)

to obtain a trace formula (for GL(2)) of the form

(5.1)  ∫ Λ_{2}^{T} K(x, x) dx = ∑_{χ} ∫ Λ_{2}^{T} K_{χ}(x, x) dx.

Our purpose now is to explain where the modified kernels are actually hiding in(5.1) and why they can not be avoided in the general theory.

Using the crucial Lemma 2.1 of Lecture II, one can easily prove (see p. 235 of [GJ]):

Lemma 5.2. For T sufficiently large (i.e. α(T) larger than the constant d_{Ω_{f}} described in Lemma 2.1 of Lecture II ),

Λ_{2}^{T} K(x, x) = ∑_{𝔬 elliptic} K_{𝔬}^{T} (x, f) + ∑_{\begin{matrix}𝔬 unipotent \\ or \\ hyperbolic \end{matrix}}K_{𝔬}^{T} (x, f) ≡ K^{T} (x, f)

Note that Lemma (5.2) implies that the truncated geometric kernel used in (5.1) is already a sum of the modified geometric kernels. Thus an alternate strategy for obtaining the trace formula

(5.3)  ∑_{𝔬} J_{𝔬}^{T} (f) = ∑_{χ} J_{χ}^{T} (f)

suggests itself: since we also have

Λ_{2}^{T} K(x, x) = ∑_{χ} Λ_{2}^{T} K_{χ}(x, x).

(the spectral analogue of the geometric formula for Λ_{2}^{T} K(x, x) afforded by Lemma (5.2)) , why not just integrate each of these expressions term by term to obtain the trace formula identity (5.3) with

J_{χ}^{T} (f) = ∫_{Z(𝔸)G(F)\G(𝔸)} Λ_{2}^{T} K_{χ}(x, x) dx?

Indeed, by Lecture II we know the left hand side of (5.3) is nicely behaved, and by the first three sections of this lecture, we know the right hand side converges too.

This strategy was in fact carried out in [GJ] for GL(2), where it turns out that terms on either side of (5.3) can be computed completely explicitly, as (linear) polynomials in T (see §6 of [GJ] and §§1 and 2 of the next lecture). Then when these expressions are plugged into the formula

tr R_{0}(f) = ∑_{𝔬∈𝔒} J_{𝔬}^{T} (f) − ∑_{χ∈𝔛\𝔛(G)} J_{χ}^{T} (f),

the terms depending on T cancel out (as they must, since the left hand side is independent of T). Thus we are left with our sought-after formula for tr R_{0}(f).

What goes wrong with this strategy in general? Well, for an arbitrary reductive group G, it would be hopeless to try and compute each term J_{𝔬}^{T} (f) and J_{χ}^{T} (f) completely explicitly; fortunately, however, it is also unnecessary. What Arthur’s more indirect strategy (involving modified spectral as well as geometric kernels) allows one to prove is that J_{χ}^{T} (f) and J_{𝔬}^{T} (f) are a priori polynomials in T (for sufficiently large α(T)). Thus we can concern ourselves only with the constant terms of these polynomials (since our goal is to make (5.3) explicit, and only terms not depending on T will survive on the right-hand side …). These constant terms ((J_{𝔬}(f) ≡ J_{𝔬}^{0}(f) and (J_{χ}(f) ≡ J_{χ}^{0}(f)) are by no means trivial to compute, but they do turn out to be feasible to treat in Arthur’s general theory, as we shall soon see.

Lecture IV. More Explicit Forms of the Trace Formula

In the first few lectures, we have described the first form of Arthur’s trace formula, which in its symmetric formulation reads

(*)  ∑ J_{𝔬}^{T} (f) = ∑_{χ} J_{χ}^{T} (f);

more suggestively,

(**)  tr(R_{0}(f)) = ∑_{𝔬} J_{𝔬}^{T} (f) − ∑_{χ∈𝔛\𝔛(G)} J_{χ}^{T} (f).

After developing this formula in [A2] and [A3], Arthur devoted the next ten years (and several hundred pages of work) to rewriting this formula in a more explicit form, suitable for applications. In particular, in order to apply (∗∗) effectively to the theory of automorphic forms, it seems essential to know a lot about the distributions J_{𝔬}^{T} (f) and J_{χ}^{T} (f). How do they depend on T? How can they be expressed explicitly in terms of weighted orbital integrals or weighted characters? How can they be factored into local distributions on the groups G_{v}?

Our purpose in the remainder of these lectures is to explain some of Arthur’s answers to these questions. But to make this task easier, we proceed in stages.

One of Arthur’s first general results is that each distribution J_{𝔬}^{T} (f) or (J_{χ}^{T} (f)) is a polynomial in T, of degree at most dim(A_{P_{0}}/Z). Another is that each “unramified” J_{𝔬}^{T} (f) (or J_{χ}^{T} (f)) has a fairly explicit representation as a weighted orbital integral (or character). To motivate these and other general results, we shall first recall the familiar example of GL(2).

1. Explicit Results for GL(2) (Geometric Side). As we already observed in general in Lecture II, the elliptic terms J_{𝔬}^{T} (f) are independent of T, and simply equal ordinary (factorizable) orbital integrals (see Proposition 2.5 of Lecture II). On the other hand, the hyperbolic and unipotent terms are non-trivial (linear) polynomials in T, explicitly described in the propositions below.

Proposition 1.1. If 𝔬 denotes a hyperbolic class with representative γ, then for α(T) sufficiently large,

J_{𝔬}^{T} (f) = (T_{1} − T_{2})m(F^{∗}\𝔸^{1}) ∫_{M(𝔸)\G(𝔸)} f(x^{−1}γx) dx

− \frac{1}{2}m(F^{∗}\𝔸^{1}) ∫_{M(𝔸)\G(𝔸)} f(x^{−1}γx)α(H(wx) + H(x)) dx

Remarks. (1) The proof of this proposition is provided by the formal computations of [GJ, pp. 237–238] now justified by the already established a priori absolute convergence of ∫ k_{𝔬}^{T} (x, f) dx. The assumption α(T) ≫ 0 is needed so as to be able to apply Lemma 4.1 of Lecture II (which in the case of GL(2) just requires α(T) > 0).

(2) In §3 of Lecture I, we saw that the unmodified hyperbolic kernel K_{𝔬} was not integrable because of the appearance of the (divergent) volume of F^{x}\𝔸^{x}; Proposition 1.1 shows that the integral of the modified kernel indeed “truncates” this volume to a finite interval of length (T_{1} − T_{2}), which appears as a linear term in T with coefficient ∫ f(x^{−1}γx) dx; however, the constant term of this polynomial no longer involves an invariant orbital integral (because of the presence of the “weight function” α(H(wx)) + H(x)).

Proposition 1.2. For the unipotent class 𝔬, and α(T) sufficiently large,

J_{𝔬}(f) = m(Z(𝔸)G(F)\G(𝔸))f(1)

+ f. p._{s=1} (ζ(F, s)) + (T_{1} − T_{2})m(F^{∗}\𝔸^{1}) ∫_{𝔸} F (y) dy

where

F (y) = ∫_{K} f(k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk is in S(𝔸),

and f. p._{s=1} (ζ(F, s)) denotes the “finite part” at s = 1 of the (meromorphic) zetafunction

ζ(F, s) = ∫_{𝔸^{x}} F (a)|a|^{s} d^{x}a

of Tate.

Remarks. (1) Iwasawa’s decomposition implies (formally) that

ζ(F, s) (at s = 1) = ∫_{𝔸^{x}} ∫_{K} f (k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk |a| d^{x}a

= ∫_{Z(𝔸)N(𝔸)\G(𝔸)} f (x^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x) dx,

the non-trivial unipotent orbital integral of f. Thus the constant term of this J_{𝔬}^{T} (f) is indeed analogous to the constant term of the hyperbolic J_{𝔬}^{T} (f): one involves a regularized (unipotent) orbital integral, and the other a weighted (hyperbolic) orbital integral.

(2) Because the left hand side of (**) is independent of T, it must be that the linear terms in T appearing in both the hyperbolic and unipotent J_{𝔬}^{T} (f) cancel out with the linear terms appearing in ∑ J_{χ}^{T} (f). Indeed one can check (using Iwasawa’s decomposition) that the linear terms in Propositions 1.1 and 1.2 (for all hyperbolic 𝔬) combine into the single term

L(T) = (T_{1} − T_{2})m(F^{∗}\𝔸^{1}) ∑_{γ∈Z(F)\M(F)} ∫_{K} ∫_{N(𝔸)} f(k^{−1}γnh) dn dh;

then this term is seen (later) to exactly cancel the linear term in T arising from ∑_{χ} J_{χ}^{T} (f); see §2 below, also p. 239 of [GJ]).

(3) The proof of Proposition 1.2 again results from formal computations—see p. 236 of [GJ]—all of which are justified now by the a priori absolute convergence of ∫ k_{𝔬}^{T} (x, f) dx. As in the hyperbolic case, the appearance of the linear term, and the “shearing off” (i.e., regularization) of the relevant orbital integral, are explained by the fact that the kernel K_{𝔬}(x, x) was modified before being integrated.

(4) The need to modify K_{𝔬} (in this unipotent case) is clarified by the exercise below.

Exercise. Show that for the unipotent class 𝔬,

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

= val_{s=1} {ζ(F, s)} + m(Z(𝔸)G(F)\G(𝔸))f(1),

with ζ(F, s) as in the proposition above. In particular, J_{𝔬}(f) = ∞ when the “principal part” of ζ(F, s), namely m(F^{∗}\𝔸^{1})F̂(0)/(s − 1), is non-zero.

Solution. Recall that

𝔬 = {1} ∪ 𝔬′,

where 𝔬′ consists of the elements

\{δ^{−1} ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) δ: t ∈ F^{∗}, δ ∈ B(F)\G(F)\}

Thus we compute

∫_{Z(𝔸)G(F)\G(𝔸)} k_{𝔬′}(x, x) dx = ∫ ∑_{t≠0} ∑_{δ∈B(F)\G(F)} (x^{−1}δ^{−1} ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) δx) dx

= ∑_{t≠0} ∫_{Z(𝔸)B(F)\G(𝔸)} f (x^{−1} ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) x) dx,

which, by Iwasawa’s decomposition, equals

∫_{K} ∫_{F^{x}\𝔸^{x}} ∑_{t∈F^{x}} f (k^{−1} ( \begin{matrix}1 & a^{−1}t \\ 0 & 1 \end{matrix} ) k) |a|^{−1} d^{x}a dk = ∫_{F^{x}\𝔸^{x}} ( ∑_{t∈F^{x}} F (at)) |a| d^{x}a
= ∫_{𝔸^{∗}} F (a)|a| d^{x}a

with F (y) as defined in Proposition 1.2.

Concluding Remark. As we shall see in the next lecture, the explicit expressions just obtained for J_{𝔬}^{T} (f) (hyperbolic or unipotent) already suffice to express these distributions as (a sum of) products of local distributions J_{v}(f_{v}); this fact will turn out to be crucial in yielding a simple form of the trace formula in Lecture V.

2. Explicit Results for GL(2) (Spectral Side). As already observed in general in Lecture III, the “elliptic” spectral terms J_{χ}^{T} (f), where χ = {(G, π)}, are independent of T; they just reduce to the distributional characters

J_{χ}^{T} (f) = tr π(f).

As for the remaining “hyperbolic” and “unipotent” χ, we have the following:

Proposition 2.1. Suppose χ = {(M, μ)} is “hyperbolic” in the sense that μ ≠ μ^{−1} (i.e., there is no 1-dimensional residual spectrum contributing to K_{χ}(x, y)). Then for T sufficiently large,

(2.1) J_{χ}^{T} (f) = (T_{1} − T_{2}) ∫_{−∞}^{∞} tr(ρ(μ, it)(f)) dt

+ ∫_{−∞}^{∞} tr(M(−it)M′(it)ρ(μ, it)(f)) dt

with the intertwining operators M(s) (and their derivatives) to be explained below.

Because the proof of this result directly generalizes to any G, we include it. By definition, for χ regular (μ ≠ μ^{−1}),

(2.2) J_{χ}^{T} (f) = ∫_{Z(𝔸)G(F)\G(𝔸)}(∑_{ϕ} ∫_{−∞}^{∞} E(x, ρ(μ, it)(f)ϕ, it)\overline{Λ^{T} E(x, ϕ, it)}) dt dx.

However, because we have already established the absolute convergence of this integral, we can interchange the orders of integration and write

(2.3)  J_{χ}^{T} (f) = ∑_{ϕ} ∫_{−∞}^{∞}(Λ^{T} E(ρ(μ, it)(f)ϕ, it), Λ^{T} E(ϕ, it)) dt.

Here we have used also the well-known fact that Λ^{T} is an orthogonal projection operator on L^{2}, and hence (ϕ_{1}, Λ^{T} ϕ_{2}) = (Λ^{T} ϕ_{1}, Λ^{T} ϕ_{2}), with (·, ·) denoting the usual inner product in L^{2}.

To continue with the proof, we obviously need a more explicit expression for the inner product of these truncated Eisenstein series. This is provided by the “Maass–Selberg relations” (generalized by Langlands to an arbitrary G in [La1]); in the present context they imply

(2.4) \begin{matrix}(Λ^{T} E(ϕ′, it), Λ^{T} E(ϕ, it)) \\ = (T_{1} − T_{2})(ϕ′, ϕ) + (M(−it)M′(it)ϕ′, ϕ) \\ + \frac{1}{2it}\{(ϕ′, M(it)ϕ)e^{2it(T_{1}−T_{2})} − (M(it)ϕ′, ϕ)e^{−2it(T_{1}−T_{2})}\}. \end{matrix}

Here M(s) = M(s, μ) is the operator

ϕ ⟶ M(s)ϕ(g) = ∫_{N(𝔸)} ϕ(wng) dn

intertwining ρ(μ, s) with ρ(μ^{w}, −s). (Like E(g, ϕ, μ, s), M(s) is initially defined only for Re(s) > 1/2, but is analytically continuable in ℂ, and holomorphic on iℝ.) Note M(−s) = M(−s, μ^{w}) maps ρ(μ^{w}, −s) to ρ(μ, s), and the inner product (ϕ_{1}, ϕ_{2}) is defined by

(ϕ_{1}, ϕ_{2}) = ∫_{Z(𝔸)M(F)\M^{1}(𝔸)} ∫_{K} ϕ, (mk)\overline{ϕ_{2}(mk)} dm dk.

Finally, M′(s) is the derivative \frac{d}{ds}M(s, μ): ρ(μ, s) ⟶ ρ(μ^{w}, −s) (defined by identifying ρ(μ, s) with ρ(μ, 0) …).

Now formula (2.4) is valid not just in our case (when μ is “regular”), but also when μ = μ^{−1}; in fact we shall require (2.4) below when we treat the “unipotent” contribution J_{χ}^{T} (f). However, when μ ≠ μ^{−1}, it is a simple matter to observe that the third (and most complicated) term on the right side of (2.4) actually vanishes. Indeed, as M(s) = M(s, μ) maps ϕ into ρ(−s, μ^{−1}), it follows that ϕ′ and M(it)ϕ transform under M^{1}(𝔸) according to distinct characters (namely, μ and μ^{−1}). Thus the inner product

(ϕ′, M(it)ϕ) = ∫ ϕ′(k)\overline{M(it)ϕ(h)} dh(∫_{Z(𝔸)M(F)\M^{1}(𝔸)} μ^{2}(m) dm)

vanishes (and so, similarly, does (M(it)ϕ′, ϕ)). So plugging (2.4) into (2.3) simply yields

(2.5) (T_{1} − T_{2}) ∫_{−∞}^{∞} tr(ρ(μ, it)(f)) dt + ∫_{−∞}^{∞} tr(M(−it)M′(it)ρ(μ, it)(f)) dt,

as required.

We now move on to the more problematic case when μ = μ^{−1}. In this case, plugging in (2.4) into (2.3) yields (in addition to the terms in (2.5)) the term

(2.6) E(T) = ∑_{ϕ} ∫_{−∞}^{∞} \{(ρ(μ, it)(f)ϕ, M(it)ϕ)\frac{e^{2it(T_{1}−T_{2})}}{2it}

− (ρ(μ, it)(f)ϕ, M(−it)ϕ)\frac{e^{−2it(T_{1}−T_{2})}}{2it} \} dt.

(Here we have used the fact that the adjoint of M(s) is M(−s).) But this still isn’t enough to describe J_{χ}^{T} (f). In this singular situation, what’s missing is the contribution from the one-dimensional residual spectrum, i.e., the contribution

J_{μ}^{T,res}(f) = μ(f) ∫_{Z(𝔸)G(F)\G(𝔸)} μ(x)Λ^{T} (\overline{μ(x)} dx

from the residual kernel

K_{μ}^{T,res}(x, y) = μ(f)μ(x)\overline{μ(y)}.

Analyzing both these functions of T (namely E(T) and J_{μ}^{T,res}(f)) yields finally:

Proposition 2.7. For α(T) ≫ 0, and χ = \{(M, μ)\} \underline{with μ = μ^{−1}},

J_{χ}^{T} (f) = (T_{1} − T_{2}) ∫_{−∞}^{∞} tr(ρ(μ, it)(f)) dt + ∫_{−∞}^{∞} tr(M(−it)M′(it)ρ(μ, it)(f)) dt,

+ \frac{1}{4} tr(M(0)ρ(μ, 0)(f)) + μ(f)m(Z(𝔸)G(F)\G(𝔸))

plus n_{1}^{∗}(T), a negligible term (for T very large).

Proof. In [GJ, pp. 239–240], a Fourier analysis argument is given to show that

E(T) = \frac{1}{4} tr(M(0)ρ(μ, 0)(f)) + r_{1}(T)

with r_{1}(T) negligible for α(T) large. On the other hand, it is a pleasant exercise to check that

J_{μ}^{T,res}(f) ⟶ μ(f)m(Z(𝔸)G(F)\G(𝔸)) as T → ∞.

Indeed,

μ(f) ∫_{Z(𝔸)G(F)\G(𝔸)} μ(x)Λ^{T} (\overline{μ(x)}) dx

= μ(f) ∫_{Z(𝔸)G(F)\G(𝔸)} μ(x)(\overline{μ}(x) − ∑_{δ∈B(F)\G(F)} \overline{μ}(x)τ̂_{B}(δx) − T)) dx

= μ(f) ∫ dx − μ(f) ∫ ∑_{δ∈B(F)\G(F)} τ̂_{B}(H(δx) − T) dx

= μ(f)m(Z(𝔸)G(F)\G(𝔸)) − μ(f) ∫_{Z(𝔸)B(F)\G(𝔸)} τ̂_{B}(H(x) − T) dx

= μ(f)m(Z(𝔸)G(F)\G(𝔸)) − n_{1}(T)

with

n_{1}(T) = μ(f)m(F^{∗}\𝔸^{1})\frac{e^{−2(T_{1}−T_{2})}}{2} ⟶ ∞ as α(T) → ∞.

So setting n_{1}^{∗}(T) = r_{1}(T) + n_{1}(T) completes the proof. □

Concluding Remark. (a) Proposition 2.1 expresses J_{χ}^{T} (f) directly as a polynomial in T when χ is regular. But as already suggested, Arthur has proven in general that J_{χ}^{T} (f) is polynomial in T. Thus it must follow that the “negligible” term n_{1}^{∗}(T) (appearing in the expression for J_{χ}^{T} (f) for singular χ in Proposition 2.7) is actually zero identically. (Indeed, it is a polynomial in T, which goes to 0 as T goes to infinity ….) On the other hand, the expression E(T) (or equivalently (6.28) in [GJ, p. 239] is definitely not a polynomial; rather E(T) includes a negligible term r_{1}(T) which must cancel the negligible (exponentially decreasing) term n_{1}(T) appearing in the expression for J_{μ}^{T,res}(f).

(b) Our computation of

J_{μ}^{T,res}(f) = μ(f) ∫_{Z(𝔸)G(F)\G(𝔸)} μ(x)Λ^{T} (\overline{μ(x)}) dx

really amounts to the computation of the inner product of two truncated “degenerate Eisenstein series” (induced from μ on P = G …). In general, the explicit description of J_{χ}^{T,res}(f) with χ “singular” will involve an inner product

(2.8)  (Λ^{T} E_{P} (ϕ′, λ′), Λ^{T} E_{P} (ϕ, λ))

where ϕ in Ind_{P (𝔸)}^{G(𝔸)} σ belongs to L^{2}(P)_{χ}, but P does not belong to P_{χ}. Thus the natural generalization of Proposition 2.7 must involve Arthur’s generalization of Langlands’ formula (valid for P ∉ P_{χ}, but like the formula for J_{μ}^{T,res}(f) giving only an asymptotic expression for (2.8); see [A9] ).

3. Results for General G. Arthur’s generalization of the above results, and in fact the GL(2) results themselves, seem much easier to digest and understand when viewed not as a long chain of unrelated results, but rather as realizations of a few fundamental phenomena and principles. Of course Arthur knowingly developed his trace formula with these thoughts in mind. To begin to explain them, let us start with the symmetry of the formula itself.

Remark 3.1. Consider the formula

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

Although at first sight the indexing sets 𝔒 and 𝔛 seem asymmetric and unrelated, they can actually be viewed as mirror images of one another. Indeed, if 𝔬 ∈ 𝔒, consider those (standard) parabolics B of G which are minimal with respect to the property that 𝔬 meets M_{B}. Then 𝔬 ∩ M_{B} is a finite union of M_{B}(F) conjugacy classes which are “elliptic” in the sense that they meet no proper parabolic subgroup of M_{B}. Moreover, if W_{0} denotes the restricted Weyl group of (G, A_{0}), then 𝔒 is in bijective correspondence with the set of W_{0}-orbits of pairs (M_{B}, c_{B}), where B is a parabolic subgroup of G, and c_{B} is an elliptic conjugacy class in M_{B}(F). Thus 𝔒 indeed corresponds naturally to 𝔛, itself defined to be the set of W_{0}-orbits of pairs (M_{B}, r_{B}), where r_{B} is an irreducible cuspidal automorphic representation of M_{B}(𝔸). This description of 𝔒 has the additional benefit that it highlights the analogy between “elliptic” classes and “cuspidal” representations. (For convenience here, we have ignored centers ….)

In light of the symmetry between 𝔒 and 𝔛 (expressed by this Remark 3.1), it is not surprising that most of the general results Arthur proves for the distributions J_{𝔬}^{T} (f) have an immediate analogue for J_{χ}^{T} (f). For example, if 𝔬 = {(M_{B}, c_{B})} is such that J_{𝔬}^{T} (f) is a polynomial of degree at most dim(A_{B}/Z), then so is J_{χ}^{T} (f), for χ = {(M_{B}, r_{B})}. Or, if 𝔬 = {(M_{B}, c_{B})} is suitably unramified that J_{𝔬}^{T} (f) is expressible as a weighted orbital integral, then the corresponding J_{χ}^{T} (f) should be expressible as a weighted average of characters. Let us explain these matters in some more detail now, and along the way bring into play some other basic principles as well.

Proposition 3.1 (see Proposition 2.3 of [A4]). For each fixed f, J_{𝔬}^{T} (f) is a polynomial in T of degree at most dim(𝔞_{P_{0}}/𝔞_{G}) (with P_{0} the minimal parabolic).

The proof has two main ingredients. First, for any parabolic P ⊃ P_{0}, and point X in 𝔞_{0}, a certain characteristic function

Γ′_{P} (H, X) = ∑_{{R: R⊃P }}(−1)^{dim(A_{R}/Z)}τ_{P}^{R}(H)τ̂_{R}(H − X)

is defined on 𝔞_{0}, with τ_{P}^{R} the characteristic function on 𝔞_{0} of the set

{H ∈ 𝔞_{0} : α(H) > 0, α ∈ Δ_{P}^{R}}

and τ̂_{R} of the set

{H ∈ 𝔞_{0} : ω̃(H) > 0, ω̃ ∈ Δ̂_{R}}.

(Here Δ̂_{R} is the dual basis in 𝔞_{R}^{∗} corresponding to the basis of coroots {α^{∨}: α ∈ Δ_{R}} in 𝔞_{R}.)

For example, for GL(2), and X = (X_{0}, −X_{0}) in 𝔞_{0}^{+}, Γ′_{B}(H, X) = τ_{B}(H)−τ̂_{B}(H − X); as a function on the one-dimensional subspace 𝔞_{0}^{G} = {(T, −T)} ≈ {T }, this is just the characteristic function of the interval [0, X_{0}].

On the other hand, for G = GL(3) and (X_{1}, X_{2}, −(X_{1} + X_{2})) = X in 𝔞_{0}^{+}, the function Γ′_{P_{0}}( · , X) (viewed on the two-dimensional space {(r_{1}, r_{2}, −(r_{1} + r_{2}))}) is just the characteristic function of the shaded region below:

Fig. 1

In general, the crucial property of Γ′_{P} (H, X) is that it has compact support, and that its integral

∫_{𝔞_{P}^{G}} Γ′_{P} (H, X) dH

is a polynomial in X of degree q = dim(A_{P} /Z). (Here 𝔞_{P}^{G} is the subspace of 𝔞_{P} annihilated by 𝔞_{G}^{∗}, so in particular, 𝔞_{P} = 𝔞_{P}^{G} ⊕ 𝔞_{G}.)

Second, certain geometric distributions J_{𝔬}^{M_{Q},T} (analogous to J_{𝔬}^{T}) are defined for each Levi subgroup M of G. Namely, for any fixed suitably regular point T_{1} in 𝔞_{0}^{+}, class 𝔬 in 𝔒 and f_{Q} in C_{c}^{∞}(Z(𝔸)\M_{Q}(𝔸)), define

J_{𝔬}^{M_{Q},T_{1}}(f_{Q}) = ∑_{i=1}^{n} J_{𝔬_{i}}^{M_{Q},T_{1}},

where 𝔬 ∩ M_{Q}(F) is the union (possibly empty) of “classes” 𝔬_{1}, … 𝔬_{n} in M_{Q}. (If 𝔬∩M_{Q}(F) = ∅, then J_{𝔬}^{M_{Q},T_{1}} ≡ 0; on the other hand, if M = G, then J_{𝔬}^{M,T_{1}} = J_{𝔬}^{T_{1}}.)

To prove Proposition 3.1, Arthur derives the formula

(3.2)  J_{𝔬}^{T} (f) = ∑_{Q⊃P_{0}} J_{𝔬}^{M_{Q},T_{1}}(f_{Q}) ∫_{𝔞_{Q}^{G}} Γ′_{Q}(H, T − T_{1}) dH,

where f_{Q} in C_{c}^{∞}(Z(𝔸)\M_{Q}(𝔸)) is explicitly determined by f (and T varies freely through T_{1} + 𝔞_{0}^{+}). Since each ∫ Γ′_{Q}(H, T − T_{1}) dH is a polynomial in T − T_{1} of degree at most dim(A_{P_{0}}/Z), it follows that the same is true of J_{𝔬}^{T} (f), as claimed. Note that J_{𝔬}^{T} (f) will be independent of T, i.e., be of degree zero, precisely when 𝔬 is elliptic, i.e., has empty intersection with every proper parabolic Q appearing in (3.2) above.

Remark 3.3. As expected, there is an analogue of Proposition 3.1 for J_{χ}^{T} (f), proved in just the same way (in keeping with the principle that the geometric and spectral terms are different facets of the same type of object). For χ in 𝔛, and M a Levi subgroup of G, one similarly defines distributions J_{χ}^{M,T} on M(𝔸) and proves (analogously) that

(3.4)  J_{χ}^{T} (f) = ∑_{Q⊃P_{0}} J_{χ}^{M_{Q},T_{1}} ∫_{𝔞_{Q}^{G}} Γ′_{Q}(H, T − T_{1}) dH

is (again) a polynomial in T of degree ≤ dim(A_{0}/Z). Interestingly, one also obtains the equality

(3.5)  ∑_{0∈𝔒} J_{𝔬}^{M,T} (f) = ∑_{χ∈𝔛} J_{χ}^{M,T} (f),

which may be viewed as (a first form of) Arthur’s trace formula “relativized” to an arbitrary Levi subgroup M_{P} of G! This relativization is reminiscent of Langlands’ use of the spectral decomposition of L^{2}(P) (P any parabolic of G) in order to describe L^{2}(G) itself, and it constitutes another of the unifying themes in Arthur’s work.

To turn next to explicit formulas for J_{𝔬}^{T} (f) (and J_{χ}^{T} (f)), we need first to recall Arthur’s notion of an unramified class of datum. For χ = {(M_{B}, r_{B})} this just means that for any pair (M_{B}, r_{B}) in χ, the only element of Ω(𝔞_{P} , 𝔞_{P}) fixing r_{B} is the trivial element. On the other hand, for 𝔬 = {(M_{B}, c_{B})}, the notion is a little less obvious.

Namely, fix a class 𝔬, and choose a parabolic subgroup P and a semisimple element γ in 𝔬 such that γ belongs to M(F) but not to the Levi subgroup of parabolic properly contained in P. Then let ∑(γ) denote the (possibly empty) set of roots α in ∑ = ∑(P, A_{P}) such that the centralizer of γ in 𝔶_{α} (the root subspace of the Lie algebra of N_{P} belonging to α) is not zero, and let A′ be the intersection of the kernels of these roots α (regarded as characters of A). Now assume that every element in 𝔬 is semi-simple, and choose a parabolic subgroup P_{1}, and an element w ∈ Ω(𝔞, 𝔞_{1}) such that A_{P_{1}} = wA′w^{−1}. Setting γ_{1} equal to wγw^{−1}, we call 𝔬 unramified if the centralizer G(γ_{1}) is contained in M_{1}.

Note that

γ = ( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) or γ = ( \begin{matrix}α & 0 & 0 \\ 0 & β & 0 \\ 0 & 0 & 1 \end{matrix} )

determines a semi-simple class in GL(2) (or GL(3)) which is unramified, with P the Borel subgroup, A′ = A_{P}, P_{1} = P, and γ_{1} = γ. On the other hand, any class in GL(3) containing a non-semisimple element, like say

γ = ( \begin{matrix}α & 0 & 0 \\ 0 & 1 & x \\ 0 & 0 & 1 \end{matrix} ) ,

is automatically ramified, and its corresponding distribution J_{𝔬}^{T} (f) will not be of the nice type described in the proposition below.

Proposition 3.6. Suppose 𝔬 is an unramified class in 𝔒, and γ_{1} is a (semisimple) element in 𝔬. Then for sufficiently large T,

J_{𝔬}^{T} (f) = m(G_{γ_{1}}(F)\G_{γ_{1}}^{1}(𝔸)) ∫_{G_{γ}(𝔸)\G(𝔸)} f(x^{−1}γ_{1}x)v(x, T) dx,

with v(x, T) (the weight function) equal to the volume of the convex hull of the points

{w^{−1}T − w^{−1}H_{P_{0}}(wx): w ∈ ⋃_{P_{2}} Ω(𝔞_{1}, 𝔞_{2})}

projected onto 𝔞_{1}/𝔞_{G}.

Let us describe v(x, T) graphically for G = GL(2) or GL(3), with γ_{1} a diagonal element (with distinct eigenvalues). We shall also sketch the proof of the proposition for GL(2), making clear how the volume v(x, T) naturally arises. N.B. We should really write v_{𝔬}(x, T) for v(x, T), since this function indeed depends on the nature of 𝔬.

For GL(2) or GL(3), and γ_{1} as just specified, v(x, T) is precisely the volume of the convex hull of the projection of

{w^{−1}T − w^{−1}H_{P_{0}}(wx): w ∈ Ω(𝔞_{P_{0}}, 𝔞_{P_{0}})}

onto 𝔞_{P_{0}}/𝔞_{G}. For GL(2), this is just the length of the line segment determined by the points (T_{1}, T_{2}) − H(x) and (T_{2}, T_{1}) − wH(wx) in 𝔞_{P_{0}} projected onto the one-dimensional space 𝔞_{P_{0}}/𝔞_{G}, i.e.,

v(x, T) = 2(T_{1} − T_{2}) − (α(H(x)) + α(H(wx))).

Thus Proposition 3.5 indeed reduces to Proposition 1.1.

Fig. 2

On the other hand, for GL(3), v(x, T) (at least for T = 0) reduces to the area of the convex set whose extreme points are indexed by the Weyl group elements {w_{1} = e, w_{2}, …, w_{6}}. Note here that

w^{−1}H_{P_{0}}(wx) = H_{P_{w}}(x),

where P_{w} denotes the (not-necessarily standard) parabolic subgroup w^{−1}P_{0}w, and H_{P_{w}}(x) is defined analogously to H_{P_{0}} using the decomposition G = P_{w}K. (Indeed wx = nm_{0}k implies w^{−1}H_{P_{0}}(wx) = w^{−1}H_{P_{0}}(m_{0}), and x = w^{−1}nm_{0}k implies H_{P_{w}}(x) = H_{P_{0}}(w^{−1}mw) = w^{−1}H_{P_{0}}(m_{0}), i.e., H_{P_{w}}(x) = w^{−1}H_{P_{0}}(wx).) Thus we could have alternately described v(x) = v(x, 0) in terms of the points

{H_{P} (x): P ∈ P(M_{0})},

where P(M_{0}) denotes the set of parabolics (not necessarily standard) in G whose Levi part equals M_{0}.

Remark 3.7. Weighted orbital integrals of the above type were first systematically studied by Arthur in the context of real groups. In [A8], Arthur studied general properties of these real weighted orbital integrals, motivated by a suggestion of Langlands that such integrals would arise in any projected general treatment of the trace formula. In particular, for matrix coefficients of discrete series representations, Arthur related ∫ f(x^{−1}γx)v(x) dx to the characters of the discrete series (whence the title of [A8]). On the other hand, Arthur’s weight functions v(x) were also interpreted in [A8] in terms of what would eventually be known as (G, M) families. We shall return to this point in a later lecture, where (G, M) families are discussed in earnest.

On the Proof of Proposition 3.6 for GL(2). Precisely because 𝔬 is unramified (in this case, a hyperbolic class with semi-simple element γ_{1}), the centralizer G(γ_{1}) is contained in M (in this case equals M) and we can express the modified kernel as

k_{𝔬}^{T} (x, T)

= ∑_{δ∈M(F)\G(F)} f(x^{−1}δ^{−1}γ_{1}δx)

− ∑_{δ∈B(F)\G(F)} ∫_{N(𝔸)} f(x^{−1}δ^{−1}γ_{1}nδx) dn τ̂_{B}(H(δx) − T)

= ∑_{B(F)\G(F)} { ∑_{η∈N(F)} f(x^{−1}δ^{−1}η^{−1}γ_{1}ηδx)

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

Thus

J_{𝔬}^{T} (x, f)

= ∫_{Z(𝔸)G(F)\G(𝔸)} k_{𝔬}^{T} (x, T) dx

= ∫_{Z(𝔸)B(F)\G(𝔸)} { ∑_{η∈N(F)} f(x^{−1}η^{−1}γ_{1}nx)

− ∫_{N(𝔸)} f(x^{−1}n^{−1}γ_{1}nx) dn τ̂_{B}(H(x) − T)} dx

(here we made the change of variables n ⟶ (γ_{1}^{−1}n^{−1}γ_{1})n in the second integral)

= ∫_{Z(𝔸)M(F)G(𝔸)} f(x^{−1}γ_{1}x)(1 − τ̂_{B}(H(x) − T)) dx,

(here we used the decomposition

Z(𝔸)B(F)\G(𝔸) ≈ N(𝔸)Z(𝔸)M(F)\G(𝔸) · N(F)\N(𝔸),

carrying out first the integration over N(F)\N(𝔸)). But now observe the following: because we know k_{𝔬}^{T} (x, T) is absolutely integrable, we also know that the last integral

A = ∫_{Z(𝔸)M(F)\G(𝔸)} f(x^{−1}γ_{1}x)(1 − τ̂_{B}(H(x) − T)) dx

converges absolutely, and equals (from the change of variables x → wx)

B = ∫ f(x^{−1}γ_{1}x)(1 − τ̂_{B}(H(wx) − T)) dx,

i.e.,

J_{𝔬}^{T} (x, f)

= A = \frac{1}{2}(A + B)

= \frac{1}{2} ∫_{Z(𝔸)M(F)\G(𝔸)} f(x^{−1}γ_{1}x)(1 − τ̂_{B}(H(x) − T) − τ̂_{B}(H(wx) − T)) dx

= ∫_{M(𝔸)\G(𝔸)} f(x^{−1}γ_{1}x)v^{∗}(x, T) dx,

with

v^{∗}(x, T) = ∫_{Z(𝔸)M(F)\G(𝔸)}(1 − τ̂_{B}(H(mx) − T)) − τ̂_{B}(H(wmx) − T)) dm,

and it remains only to prove:

Claim. Set v_{T}^{∗} (x) = 1 − τ̂_{B}(H(x) − T) − τ̂_{B}(H(wx) − T). Then

v^{∗}(x, T) = ∫_{Z(𝔸)M(F)\M(𝔸)} v_{T}^{∗} (mx) dm = Arthur’s weight factor v(x, T).

Proof. Note that v_{T}^{∗} (x) is identically zero unless

(a) Both τ̂_{B}(H(x) − T) and τ̂_{B}(H(wx) − T) are 1 (in which case v_{T}^{∗} (x) = −1); or

(b) Both are zero (in which case v_{T}^{∗} (x) = 1). But the first possibility for v_{T}^{∗} (mx) ≠ 0 implies that

α(H(mx)) > α(T) \underline{and} α(H(wmx)) > α(T),

which (for α(T) sufficiently large …) implies (by Lemma 4.1 of Lecture II) that w ∈ B(F), an obvious contradiction. On the other hand, it is straightforward to check that the second possibility for v_{T}^{∗} (mx) ≠ 0 implies

α(H(mx)) − (T_{1} − T_{2}) < α(H(m)) < (T_{1} − T_{2}) − α(H(x)).

Thus the decomposition

Z(𝔸)M(F)\M(𝔸) = \{( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) : a ∈ F^{x}\𝔸^{1}\} · \{( \begin{matrix}e^{t} & 0 \\ 0 & 1 \end{matrix} )\}

∫ v_{T}^{∗} (mx) dx = ∫_{F^{x}\𝔸^{1}} d^{x}a ∫_{α(H(wx))−(T_{1}−T_{2})}^{(T_{1}−T_{2})−α(H(x))} dt
= m(F^{x}\𝔸^{1})v(x, T).
□

Remark 3.8. The analogue of Proposition 3.6 for a spectral unramified distribution J_{χ}^{T} (f) is also a “straightforward” generalization of the GL(2) situation (Proposition 2.1), namely, we have:

Proposition 3.9. For χ = {(M, r)} unramified,

J_{χ}^{T} (f) = ∑_{P ∈P_{χ}} \frac{1}{n(A_{p})} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} tr(M_{P}^{T} (σ)_{χ}ρ(σ, λ)(f)) dλ,

with M_{P}^{T} (σ)_{χ} the operator on Ind_{P} (σ, λ) defined by

c_{P} lim_{ζ→0} ∑_{P_{2}∈P_{χ}} ∑_{w∈Ω(𝔞_{P_{1}} ,𝔞_{P_{2}} )} \frac{e^{wζ(T)}M(w, 0, σ)^{−1}M(w, ζ, σ)}{∏_{α∈Δ_{P_{2}}}(wζ)(α^{∨})}

and M(w, ζ, σ) the operator intertwining Ind_{P_{1}}^{G} σ| |^{λ+ζ} with Ind_{P_{2}}^{G} σ^{w}| |^{w(λ+ζ)}.

As suggested in §2, the crucial point of the proof is Langlands’ formula for the inner product of truncated Eisenstein series, and since this is valid for general G (for Eisenstein series E_{P} with P ∈ P_{χ}), the proof of Proposition 2.1 generalizes directly. The interesting point here is that the integrals appearing in the description of J_{χ}^{T} (f) above really should be viewed as weighted character sums. In fact, using the notion of (G, M) families, Arthur eventually views both M_{P}^{T} (σ) and the weight functions v(x) as special examples of functions constructed out of similar (G, M) families; we shall return to this point in Lecture VII.

(New) References

[A8] Arthur, J., The characters of discrete series as orbital integrals, Invent. Math. 32 (1976), 205–261.

[A9] , On the inner product of truncated Eisenstein series, Duke. Math. J. 49 (1982), no. 1, 35–70.

Lecture V. Simple Forms of the Trace Formula

In this lecture and the next, we explain (at least for GL(2)) how the trace formula can be molded into a simpler form, and then used to obtain dramatic results in representation theory, number theory, and the theory of automorphic forms. In addition to providing a welcome respite from the general theory, this detour will provide impetus and direction for the further discussion of Arthur’s development of a general trace formula.

Roughly speaking, the trace formula takes on a simpler form when more restrictive hypotheses are put on the test functions f. The more restrictive the hypotheses, the easier it is to establish the corresponding “simple trace formula”; but the less restrictive the hypotheses, the more effective it is to apply the resulting trace formula to the theory of automorphic forms.

For example, the “very simple” trace formula of Deligne-Kazhdan (described in §3 below) has nice applications to local representation theory, but cannot give complete results on the functorial lifting of automorphic forms, and cannot give applications to the computation of Tamagawa numbers. For such applications one seems to need the “simple trace formula of Arthur”, described in §2 below (in the context of GL(2)).

We start by showing (in §1) that each term on the right side of the formula

tr(R_{0}(f)) = ∑_{𝔬} J_{𝔬}^{T} (f) − ∑_{χ∉𝔛(G)} J_{χ}^{T} (f)

may be expressed as a finite sum of products of local distributions on each G_{v}. Then we show (in §2) that for specially chosen f = ∏_{v} f_{v}, sufficiently many of these local distributions vanish to ensure that the resulting trace formula reduces to the simple formula

tr R_{d}(f) = m(Z(𝔸)G(F)\G(𝔸))f(1) + ∑_{𝔬 ; elliptic} J_{𝔬}(f)

(as in the case of compact quotient).

Henceforth, we assume

f = ∏_{v} f_{v}

is such that for all v, f_{v} ∈ C_{c}^{∞}(Z_{v}\G_{v}), and for almost all v, f_{v} is the characteristic function of Z_{v}K_{v}.

1. Factorization into Local Distributions. We start with the geometric terms J_{𝔬}(f). Fix f = Πf_{v}, and let S_{f} denote the finite set of places of F outside of which f_{v} is the characteristic function of Z_{v}K_{v}.

Proposition 1.1. Suppose γ is hyperbolic and belongs to 𝔬. Then J_{𝔬}(f) (the constant term of the polynomial J_{𝔬}^{T} (f)) is expressible as the sum (over v in S_{f}) of the products

c( ∏_{w≠v} ∫_{M_{w}\G_{w}} f_{w}(g^{−1}γg) dg) ∫_{M_{v}\G_{v}} f_{v}(g^{−1}γg)v_{v}(g) dg.

Here v_{v}(g) = α(H_{v}(g) + H_{v}(wg)) is the local weight function.

Proof. It is easy to check that for g in G(𝔸),

v(g) = ∑_{v} v_{v}(g_{v}),

with v_{v}(g_{v}) ≡ 0 on Z_{v}K_{v}. Thus, Proposition 1.1 of Lecture IV immediately implies

J_{𝔬}(f) = c ∑_{v} ∫_{M(𝔸)\G(𝔸)} f(g^{−1}γg)v_{v}(g) dg

= ∑_{v} c( ∏_{w≠v} ∫_{M_{w}\G_{w}} f_{w}(g^{−1}γg) dg) ∫_{M_{v}\G_{v}} f_{v}(g^{−1}γg)v_{v}(g) dg.

But if v ∉ S_{f}, the second integrand is identically zero. Thus the proposition is clear. □

The factorization of the unipotent term is a bit more complicated. Recall from Proposition 1.2 of Lecture IV that the constant term of the (non-trivial part of the) unipotent term J_{𝔬}^{T} (f) is computed by subtracting off from

ζ(F, s) = ∫_{Z^{x}} ∫_{K} f(k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk|a|^{s} d^{x}a

its principal part at s = 1, and then setting s = 1.

So first rewrite this last integral as L(s, 1_{F})θ(s), where

θ(s) = \frac{1}{L(s, 1_{F})} ∫∫ f(k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk|a|^{s} d^{x}a.

By Tate’s theory, θ(s) is holomorphic at s = 1, whereas

L(s, 1_{F}) = \frac{λ_{−1}}{s − 1} + λ_{0} + · · · .

Therefore, the constant term J_{𝔬}(f) we need to compute is λ_{−1}θ′(1) + λ_{0}θ(1), and arguing as in [GJ, pp. 242–243], we get:

Proposition 1.2. With S_{f} as before, and 𝔬 unipotent,

J_{𝔬}(f) = λ_{0} ∏_{v}(\frac{1}{L(1, 1_{v})} ∫_{Z_{v}N_{v}\G_{v}} f_{v}(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg)

+ λ_{−1} ∑_{u∈S_{f}} ∏_{v≠u}(\frac{1}{L(1, 1_{v})} ∫_{Z_{v}N_{v}\G_{v}}(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg_{v}) × \frac{d}{ds} |_{s=1}

\frac{1}{L(s, 1_{u})} ∫∫ f_{u}(k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk |a|^{s} d^{x}a

+ m(Z(𝔸)G(F)\G(𝔸)f(1).

Remarks. (1) Each of the local unipotent orbital integrals

∫_{Z_{v}N_{v}\G_{v}} f_{v}(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg = ζ(F_{v}, 1)

converges, and for almost every v equals (1 − N_{v}^{−1})^{−1}. It is the appearance of the “convergence factors” L(1, 1_{v})^{−1} which makes possible the convergence of the product of these local integrals in Proposition 1.2. (The divergence of the global unipotent orbital integral

∫ f(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg = ∏_{v} ∫ f_{v}(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg

was of course the reason for introducing the modified J_{𝔬}^{T} (f) in the first place …).

(2) If u ∉ S_{f}, then

\frac{1}{L(s, 1_{u})} ∫∫ f_{u}(k^{−1} ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) k) dk|a|^{s} d^{x}a = \frac{1}{L(s, 1_{u})} · ζ(1_{O_{v}}, s) = 1.

and its derivative is zero; that’s why we need only sum over u ∉ S_{F} in Proposition 1.2.

We turn now to the spectral contributions. Here we need to recall some crucial facts about normalized intertwining operators.

Recall the operator

(M(s)ϕ)(g) = ∫_{N(𝔸)} ϕ(wng) dn (Re(s) > \frac{1}{2})

intertwining ρ(μ, s)with ρ(μ^{w}, −s). This is the operator—or rather its analytic continuation to iℝ—which appears in our description of the spectral “constant term” J_{χ}(f)(χ = (M, μ)). The problem is that in analytically continuing M(s) we lose its Euler product factorization (just as we lose the Euler product for ζ(s) in analytically continuing it to the left of Re(s) = 1). It is to restore this factorization that one needs the normalized intertwining operators R(s) recalled below.

For a fixed additive character ψ = ∏_{v} ψ_{v} of F \𝔸, and v any place of F, set

m_{v}(s, μ_{v}, ψ_{v}) = \frac{L(s, μ_{v}^{2})}{L(s + 1, μ_{v}^{2})ε(s, μ_{v}^{2}, ψ_{v})}.

Then define local normalized intertwining operators

R(s, μ_{v}) : ρ(s, μ_{v}) ⟶ ρ(−s, μ_{v}^{w})

by the equality

M(s, μ_{v}) = m(s, μ_{v}, ψ_{v})R(s, μ_{v})

(where M(s, μ_{v})ϕ is just defined by the local integral ∫_{N_{v}} ϕ(wng) dn, convergent for Re(s) ≫ 0). The crucial property of this normalized operator R(s, μ_{v}) is that it takes the (normalized) K_{v}-fixed vector ϕ_{v}^{0} in ρ(s, μ_{v}) (for μ_{v} unramified) to the unique normalized K_{v}-fixed vector in ρ(−s, μ_{v}^{−1}); moreover, as an operator valued function of s, R(s, μ_{v}) is holomorphic in Re(s) > −\frac{1}{2}. Thus we can define, for ϕ = ∏ ϕ_{v} in ρ(s, μ),

R(s, μ)ϕ = ∏_{v} R(s_{v}, μ_{v})ϕ_{v}.

Since R(s, μ_{v})(ϕ_{v}) = ϕ_{v}^{0} almost everywhere, this product makes perfect sense; moreover, R(s, μ) is holomorphic in Re(s) > −\frac{1}{2}. But clearly (at least for Re(s) ≫ 0),

M(s, μ) = m(s, μ)R(s, μ),

where

m(s, μ) = ∏_{v} m_{v}(s, μ_{v}, ψ)v).

So since m(s, μ) is meromorphic, we indeed obtain the analytic continuation of M(s, μ) to iℝ with its “Euler product” factorization intact. N.B. By the (global) functional equation of Hecke L-functions,

m(s, μ) = \frac{L(s, μ^{2})}{L(s + 1, μ^{2})ε(s, μ^{2}, ψ)} = \frac{L(1 − s, μ^{−2})}{L(s + 1, μ^{2})}

Proposition 1.3. For χ “unramified”, the (constant term of the) spectral contribution J_{χ}(f) equals

∫_{−∞}^{∞} \frac{m′(it, μ)}{m(it, μ)} tr(ρ(it, μ)(f)) dt

+ ∑_{u∈S_{f}} ∫_{−∞}^{∞}(∏_{v≠u} tr(ρ(it, μ_{v})(f_{v})) tr(R_{u}(it, μ_{u})^{−1}R′_{u}(it, μ_{u})ρ(μ, it)(f_{u})) dt.

In case χ is “ramified” (μ^{2} ≡ 1), there are the additional terms

(1.4)  tr(M(0)ρ(μ, 0)(f))

and

μ(f)m(Z(𝔸)G(F)\G(𝔸)).

Proof. Straightforward (see [GJ, p. 243]). □

2. Simplifications of the Formula. A first kind of simplification comes form making the relatively mild assumption on

f = ∏ f_{v}

that its local hyperbolic orbital integrals

(*)  ∫_{M_{v}\G_{v}} f(g^{−1} ( \begin{matrix}a_{v} & 0 \\ 0 & 1 \end{matrix} ) g) d^{∗}g ≡ 0

for at least two place v = v_{1}, v_{2}.

Proposition 2.1. With f satisfying the above assumption (*), we have

tr R_{0}(f) = m(Z(𝔸)\G(𝔸))f(1) + ∑_{𝔬 elliptic} J_{𝔬}(f) − ∑_{μ^{2}=1} μ(f)m(Z(𝔸)G(F)\G(𝔸))

Proof. First consider the spectral contributions J_{χ}(f), with χ = {(M, μ)}. If v_{1} is a place where (*) holds, then tr ρ(it, μ_{v_{1}})(f_{v}) = 0 (by a well known computation of the trace of induced representations). Thus the first term appearing in Proposition 1.3 immediately disappears. But if (*) also holds at a second place v_{2}, then it is clear that the second term there must also always vanish. On the other hand, if μ^{2} = 1, then M(0, μ) intertwines the irreducible representation ρ(0, μ) with itself; hence M(0, μ) must be a scalar operator. In particular, tr(M(0)ρ(0, μ)(f)) = λ tr(ρ(0, μ)(f)) = 0 by our assumption on f, and we conclude from Proposition 1.3 that the full spectral contribution ∑_{χ} = (M, μ)J_{χ}(f) reduces to (∑_{μ^{2}=1} μ(f))m((Z(𝔸)G(F)\G(𝔸)).

Now we consider the possible geometric contributions J_{𝔬}(f), following Propositions 1.1 and 1.2. For the hyperbolic contributions, it is again obvious that if assumption (*) holds at two places, then each J_{𝔬}(f) is zero (just as for the unramified spectral forms). On the other hand, for the unipotent contribution, it suffices to make the following observation:

For any local f_{v} in C_{c}^{∞}(Z_{v}\G_{v}),

∫_{Z_{v}N_{v}\G_{v}} f_{v}(g^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg = lim_{a→1} |1 − a^{−1}| ∫_{M_{v}\G_{v}} f_{v}(g^{−1} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) dg.

This is a simple yet important identity, which we encourage the reader to verify (using Iwasawa’s decomposition and an appropriate change of variables in N). It implies that the local unipotent orbital integral of f_{v} vanishes as soon as its hyperbolic orbital integrals vanish. In particular, as soon as (*) holds for two places, it follows that all of the unipotent term J_{𝔬}(f) vanishes, except for the term involving f(1). □

In the next lecture, we shall explain how this simplification of the trace formula is used to prove the Jacquet–Langlands correspondence between automorphic representations between a quaternion algebra and GL(2), and to show that the corresponding Tamagawa numbers are equal.

3. The “Simple” Trace Formula. In the early 1980’s, Deligne and Kazhdan introduced a remarkably simple trick into the trace formula repertoire, the so-called “simple trace formula” (see [BDKV]). As we shall see in the next lecture, it yields powerful local results on functorial lifting, with surprisingly little work.

Theorem 3.1 (The Simple Trace Formula of Deligne-Kazhdan). Suppose f = ∏_{v} f_{v} in C_{c}^{∞}(Z(𝔸)\G(𝔸)) satisfies the following two properties:

(i) At one place v = v_{1}, f_{v_{1}} is the matrix coefficient of a supercuspidal representation of G_{v_{1}}; and

(ii) at a second place v = v_{2}, f_{v_{2}} is supported on the set of regular elliptic

elements of G_{v_{2}}. Then

(a) R(f) has its image in L_{0}^{2}(Z(F)G(F)\G(𝔸)), hence is of trace class, with

tr R(f) = ∑_{π cuspidal} tr π(f); and

(b) we also have

tr R(f) = ∑_{{γ} ; regular ; elliptic} m(Z(𝔸)G_{γ}(F)\G_{γ}(𝔸) ∫_{G_{γ}\G(𝔸)} f(x^{−1}γx) dx

Remarks. (1) The condition we really need in (i) is that f_{v_{1}} is a supercusp form in the sense of Harish-Chandra, i.e., (like the matrix coefficient of any supercuspidal representation),

∫_{N_{v_{1}}} f_{v_{1}}(gnh) dn = 0 for all g, h, in G_{v}.

(2) Both the theorem, and its proof below, hold more generally for any G; see [Ro1], §1, for example.

(3) This simple trace formula is indeed less subtle than the simple form of Arthur’s trace formula discussed in the last section. In particular, note that the distribution f → m(Z(F)G(F)\G(𝔸))f(1) disappears entirely from this simpler form of the trace formula (and hence, for example, no application to Tamagawa numbers is possible …). It is just this crudeness, however, that makes the formula much easier to prove!

Proof. For (a), it suffices to show that R(f) has image in L_{0}^{2}(Z(F)G(F)\G(𝔸) for then the theorem of Gelfand and Piatetski-Shapiro (see [GPS] and [Go2]) implies R(f) is of trace class). To check that R(f)ϕ is a cusp form for any ϕ we simply compute:

∫_{N(F)\N(𝔸)} R(f)ϕ(nx) dx = ∫_{N(F)\N(𝔸)}(∫_{Z(𝔸)\G(𝔸)} ∑_{v∈N(F)} f(x^{−1}nvg)ϕ(g) dg) dn

= ∫_{Z(𝔸)N(F)\G(𝔸)}(∫_{N(𝔸)} f(x^{−1}ng) dn)ϕ(g) dg = 0

since

∫_{N(𝔸)} f(x^{−1}ng) dn = ∏_{v} ∫_{N_{v}} f_{v}(x_{v}^{−1}ng_{v}) dn = 0

by our assumption (i) on f_{v_{1}}.

To prove (b), note that a γ in G(F) is regular elliptic as soon as it is regular elliptic in G_{v} = G(F_{v}) for some place v. But f(x^{−1}γx) ≠ 0 implies {supp(f)} ∩ {Orbit of γ} ≠ 0, which in turn implies {supp(f_{v_{2}})} ∩ {Orbit of γ} ≠ 0. So by hypothesis (ii), γ is regular elliptic in G_{v_{2}}, i.e.,

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

and conclusion (b) follows. □

Concluding Remark. Implicit in the proof above is the fact that K_{f}(x, x) is an absolutely integrable smooth function on Z(𝔸)G(F)\G(𝔸), and hence tr(R(f)) (= tr(R_{0}(f))) is given by its integral. In fact (3.2) implies K_{f}(x, x) is compactly supported on Z(𝔸)G(F)\G(𝔸) (as we saw in Lecture II). Interestingly enough, if we drop the hypothesis (ii), so that (3.2) need no longer hold, it still follows (from (i) above) that K_{f}(x, x) is absolutely integrable (in fact, rapidly decreasing), and that tr R(f) = ∫ K_{f}(x, x) dx; for a proof, see [Ro1, §1].

(New) References

[BDKV] Bernstein, J., Deligne, P., Kazhdan, D, and Vigneras, M.-F., Representations des groupes sur un corps local, Hermann, Paris, 1984.

[Clo] Clozel, L., Invariant harmonic analysis on the Schwartz space of a reductive p-adic group, in Harmonic Analysis on Reductive Groups, edited by W. Barker and P. Sally, Progress in Mathematics, vol. 101, Birkhäuser, Boston, 1991, pp. 101–121.

[GPS] Gelfand, I. M., and Piatetski-Shapiro, I. I., Automorphic functions and representation theory, Trudy Moskov. Mat. Obš č. 12 (1963), 389–412; Trans. Moscow Math. Soc. 12 (1963), 438–464.

[Go2] Godement, R., The Spectral Decomposition of Cusp Forms, Proc. Sympos. Pure Math., vol. IX, American Mathematical Society, Providence, RI, 1966, pp. 225–234.

[Ro1] Rogawski, J., Representations of GL(n) and Division Algebras on a p-adic Field, Duke. Math. J. 50 (1983), no. 1, 161–196.

Lecture VI. Applications of the Trace Formula

Let G = GL(2), and let G′ be the multiplicative group of a division quaternion algebra D over F.

In this lecture, we shall explain how the simple forms of the trace formula are used to prove the following three results:

(1) The Tamagawa number of G′ equals the Tamagawa number of G;

(2) There is a bijection (the “Jacquet–Langlands correspondence”) between the automorphic π′ on G′ (which are not one-dimensional) and the automorphic cuspidal π of G (such that π_{v} is square integrable for each place v of F ramified in D); and

(3) Given any finite set S of finite places v of F, and square-integrable π_{v} on G_{v} for v ∈ S, there exists a cuspidal π on G such that (π)_{v} ≅ π_{v} for all v ∈ S.

1. Tamagawa Numbers. At the end of the 1950’s, Weil conjectured that the Tamagawa number of any simply connected semi-simple group G equals one. At the end of the 1960’s Jacquet–Langlands proposed a two step program for proving this:

(1) Prove first that

τ(G_{0}) = 1

for G_{0} the quasi-split inner form of G, using the Eisenstein series method introduced by Langlands for split groups (see [La3] and [Lai]); then

(2) Use the trace formula to prove that

τ(G) = τ(G_{0}).

This second step was first carried out in §16 of [JL] (for GL_{2} in place of SL_{2}), and then recently generalized to arbitrary G by Kottwitz to prove Weil’s conjecture in general (see [Kot]).

We should stress that—in deriving a simple trace formula for an arbitrary semisimple quasi-split G—Kottwitz had to appeal not only to all of Arthur’s work on the trace formula through 1988, but also to his own earlier works on the stable form of the trace formula. By specializing Kottwitz’s argument below to the case of GL(2) (in place of SL(2)), we manage to avoid both these bodies of work.

We start by recalling the notion of Tamagawa measures for our G and G′.

Fix a non-trivial character ψ = ∏ ψ_{v} of F \𝔸. For each place v of F, ψ_{v} determines a Fourier transform on A_{v} = M_{2}(F_{v}) or D_{v}, hence also a self-dual Haar measure dy on A_{v}. The measure dy then determines a Haar measure dy/∥y∥ on A_{v}^{x}, where ∥y∥ is the module homomorphism from A_{v} to ℝ_{+}^{x}. Hence the choice of ψ simultaneously determines Haar measures on each G_{v} and G′_{v}; similarly, Haar measures are determined on F_{v} = Z_{v} ≃ Z′_{v}. Such measures are called Tamagawa measures, locally and globally. The resulting measure of Z(𝔸)G(F)\G(𝔸) is independent of ψ and called the Tamagawa number τ(G) of G, i.e.,

τ(G) = m(Z(𝔸)G(F)\G(𝔸)).

Similarly,

τ(G′) = m(Z′(𝔸)G′(F)\G′(𝔸)).

The center Z′ may of course be identified with the center Z ≃ F^{x} of GL(2).

Theorem 1.1. The Tamagawa numbers of G and G′ are equal.

We start the proof with a definition.

Suppose v in F is ramified in D, i.e., D_{v} = D ⊗ F_{v} is a division quaternion algebra over F_{v}. Then each quadratic extension L_{v} of F_{v} can be regarded simultaneously as a Cartan subgroup T′_{v} = T′_{L_{v}} of G′_{v} and as an (elliptic) Cartan subgroup T_{v} = T_{L_{v}} of G_{v} = GL_{2}(F_{v}).

Definition 1.2. We say a function f_{v} in C_{c}^{∞}(Z_{v}\G_{v}) matches f′_{v} in C_{c}^{∞}(Z′_{v}\G′_{v}) (and write f_{v} ∼ f′_{v}) if:

(i) f_{v}(1) = f′_{v}(1); (ii) the regular hyperbolic orbital integrals

∫_{M_{v}\G_{v}} f_{v}(g^{−1} ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ) g) dg

vanish identically; and (iii) for corresponding tori T_{v} and T′_{v} and t ∼ t′,

∫_{T_{v}\G_{v}} f_{v}(g^{−1}tg) dg = ∫_{T′_{v}\G′_{v}} f′_{v}(g^{−1}t′g) d′g.

N.B. The Haar measures on G_{v} and G′_{v} are (simultaneously) normalized by way of the Tamagawa measures recalled above; the measures on T_{v} and T′_{v} are fixed by a choice of Haar measure on L_{v}.

Fact 1 (Local Harmonic Analysis on G). Given any f′_{v} in C_{c}^{∞}(Z′_{v}\G′_{v}), there exist (infinitely many …) f_{v} in C_{c}^{∞}(Z_{v}\G_{v}) which match f′_{v} in the above sense.

This crucial fact can be proved using a characterization of the orbital integrals on GL_{2} à la “Shalika germs”; see §6 of [La2]

Using Fact 1, we can relate Tamagawa numbers to traces, and then prove Theorem 1.1.

Proposition 1.3. Let S_{D} denote the set of v in F ramified in D. Given f′ = ∏_{v} f′_{v} in C_{c}^{∞}(Z′(𝔸)\G′(𝔸)), suppose f = ∏ f_{v} in C_{c}^{∞}(Z(𝔸)\G(𝔸)) is such that for all v in S_{D}, f_{v} ∼ f′_{v} (in the sense of Definition 1.2 ), and for all v ∉ S_{D}, f_{v} = f′_{v} (via the natural isomorphism of G_{v} and G′_{v}). (In this case, we say f “globally matches” f′, and write f ∼ f′). Then

(1.4)  tr R_{0}(f) − tr R′_{0}(f′) = {τ(g) − τ(G′)}(f(1) + ∑_{μ^{2}=1} μ(f)),

with R′_{0} the representation of G′(𝔸) in the subspace of L^{2}(Z(𝔸)G′(F)\G′(𝔸)) orthogonal to all one-dimensional invariant subspaces.

Assuming this proposition (for the moment), we give the:

Proof of Theorem 1.1. If we rewrite (1.4) as

(1.5) tr R_{0}(f) − tr R′_{0}(f′) − {τ(G) − τ(G′)} ∑_{μ^{2}=1} μ(f) = {τ(G) − τ(G′)}(f(1),

then it clearly suffices to prove that both sides of this equation—valid for all f ∼ f′ as above—equal zero. For this, we follow an argument first explicitly introduced by Langlands in [La2].

Fix a place v_{0} outside S_{D}, and consider the Hecke algebra H_{v_{0}}(G_{v_{0}}, K_{v_{0}}). If we fix all components of

f′ = ∏_{v} f′_{v}

except the v_{0}-component, which we let vary through H_{v_{0}}, then (1.5) reads

(1.6)  ∑_{j} c_{j} tr(π_{j})_{v_{0}}(f_{v_{0}}) − ∑_{μ^{2}=1} c_{μ}μ_{v_{0}}(f_{v_{0}}) = c_{0}{τ(G) − τ(G′)}f_{v_{0}}(1),

where the π_{jv_{0}} (and μ_{v_{0}}) are unramified unitary representations of G_{v_{0}}. But the distribution f_{v_{0}} → f_{v_{0}}(1) is given in terms of tr π_{v_{0}}(f_{v_{0}}) (for all unramified tempered π_{v_{0}}) by integration against Plancherel’s measure, which is continuous in the obvious sense. Similarly, the left-hand side of (1.6) defines a discrete measure (on this same unramified dual). Thus it follows from (the uniqueness part of) the Riesz representation theorem for measures that both sides of (1.6) must be zero. □

Corollary (of Proposition 1.3 and the Proof of Theorem 1.1). For matching f and f′ on G(𝔸) and G′(𝔸),

tr R_{0}(f) = tr R′_{0}(f′),

where R′_{0} is the representation of G′(𝔸) in the subspace of L^{2}(Z(𝔸)G′(F)\G′(𝔸)) orthogonal to all one-dimensional invariant subspaces.

It remains now to complete the:

Proof of Proposition 1.3. Because |S_{D}| ≥ 2, our assumption on f (that it matches f′) implies that f_{v} has vanishing (regular) hyperbolic orbital integrals for at least two places v_{1} and v_{2}. Thus by Proposition 2.1 of the last lecture (Lecture V)

(1.7) tr R_{0}(f) = m(Z(𝔸)G(F)\G(𝔸))f(1)

+ ∑_{γ regular ; elliptic} m(Z(𝔸)G_{γ}(F)\G_{γ}(𝔸) ∫_{G_{γ}(𝔸)\G_{γ}(𝔸)} f(x^{−1}γx) dx

− ∑_{μ^{2}=1} μ(f)m(Z(𝔸)G(F)\G(𝔸)).

On the other hand, for G′, the trace formula for compact quotient in Lecture I implies

tr R′_{0}(f′) = m(Z(𝔸)G′(F)\G(𝔸))f′(1)

+ ∑_{{γ}inZ′(F)\G′(F)} m(Z(𝔸)G′_{γ}(F)\G′_{γ}(𝔸) ∫_{G′_{γ}(𝔸)\G′(𝔸)} f′(x^{−1}γx) dx

− ∑_{μ^{2}=1} μ(f′)m(Z(𝔸)G′(F)\G′(𝔸))

So to prove the proposition, it remains to check that: (i) the regular elliptic orbital integral terms match up on G and G′; and (ii) μ(f) = μ(f′) for each character μ.

As for (i), let {L} run through a set of representatives for the classes of quadratic extensions L of F which don’t split at any v ∈ S_{D}. Then each integral

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

in (1.7) is of the form

(*)  m(𝔸^{x}L^{x}\L^{x}(𝔸)) ∫_{L^{x}(𝔸)\G(𝔸)} f(g^{−1}γg) dg

for some γ ≠ 1 in F^{x}\L^{x}. (Note that if γ belongs to a quadratic extension L which splits at some v ∈ S_{D}, then our hypothesis on f implies

∫_{T_{Lv} \G_{v}} f(g_{v}^{−1}γg_{v}) dg_{v} = 0,

i.e., the orbital integrals for such γ don’t appear in the expression for tr R_{0}(f).) On the other hand, each regular orbital integral

m(Z(𝔸)G′_{γ}(𝔸)\G′_{γ}(𝔸)) ∫_{G′_{γ}(𝔸)\G′(𝔸)} f′(x^{−1}γx) dx

appearing in the expression for tr R′_{0}(f′) is also of the form (*), since quadratic L which don’t split at v ∈ S_{D} are precisely the quadratic extensions of F embeddable in D. Thus the regular elliptic orbital integrals indeed match up term by term.

As for (ii), let us verify that

μ_{v}(f_{v}) = μ_{v}(f′_{v})

for each place v of F. For v ∉ S_{D} this is a tautology. So let us assume v ∈ S_{D}. By Weyl’s integration formula for Z′_{v}\G′_{v} we compute

\begin{aligned}μ_{v}(f′_{v}) & ≅ ∫_{Z_{v}\G′_{v}} f′_{v}(g)μ_{v}(det g) dg \\ & = ∑_{\{L_{v}\}} \frac{1}{2} ∫_{Z_{v}\T′_{Lv}} δ(t)μ(det t)(∫_{T′_{Lv} \G′_{v}} f′_{v}(g^{−1}tg) dg) dt_{v} \end{aligned}

where {L_{v}} indexes the Cartan subgroup T′_{L_{v}}, and

δ(t) =| \frac{(a_{1} − a_{2})^{2}}{a_{1}a_{2}} |

if t has eigenvalues a_{1} and a_{2}. But because f_{v} matches f′_{v}, this last sum equals

∑_{\{L_{v}\}} \frac{1}{2} ∫_{Z_{v}\T_{Lv}} δ(t)μ(det t)(∫_{T_{Lv} \G_{v}} f_{v}(g^{−1}tg) dg) dt_{v} = ∫_{Z_{v}\G_{v}} f_{v}(g)μ_{v}(g) dg

(since the orbital integrals of f_{v} vanish off the elliptic Cartans). Thus μ_{v}(f′_{v}) = μ(f_{v}), as required, and the proof of Proposition 1.3 (and hence Theorem 1.1) is complete. □

2. The Jacquet–Langlands Correspondence. Let A_{0}(G) denote the collection of irreducible invariant subspaces V_{π} of R_{0}(g) in L_{0}^{2}(Z(𝔸)G(F)\G(𝔸)). According to the “multiplicity one” result of [JL], proved only using the theory of Whittaker models, each irreducible unitary representation

π = ⊗π_{v}

of G(𝔸) is realizable at most once in L_{0}^{2}. Thus the (distinct) spaces V_{π} in A_{0}(G) may be confused with the set of (classes of) irreducible automorphic cuspidal representations of GL(2). Moreover, “strong multiplicity one” (proved using “only” the theory of L-functions) implies that V_{π} = V_{π′} as soon as π_{v} and π′_{v} are equivalent for almost all v.

Now consider the group G′ = D^{x}, and the collection A_{0}(G′) of irreducibly invariant subspaces of R′_{0}(g) acting in the subspace of L^{2}(Z(𝔸)G′(F)\G(𝔸)) orthogonal to all the one-dimensional invariant subspaces. Then the trace formula implies that “multiplicity one” and “strong multiplicity one” hold in A_{0}(G′) as well (even though Whittaker models disappear in this setting, and the theory of L-functions cannot be developed à la Hecke …). Indeed, this is a corollary of the following Jacquet–Langlands correspondence between G′ and G:

Theorem 2.1. There exists a 1–1 mapping

V_{π′} ⟶ V_{π}

from A_{0}(G′) to A_{0}(G), with the property that π_{v} ≡ π′_{v} for all v ∉ S_{D}; the image consists of all those π in A_{0}(G) with the property that π_{v} is square-integrable (mod Z) for all v ∈ S_{D}.

Remark. A weaker form of this correspondence asserts the existence of a map between the representations π′ and π above; i.e., to each π′ on G′ realizable in some G′ on A_{0}(G) there exists a π in A_{0}(G) such that π′_{v} ≡ π_{v} for all v ∉ S_{D}. As noted in Lecture I, this can be proved using L-functions but the argument does not yield a characterization of the image, nor the multiplicity one results for G′.

The trace formula proof of Theorem 2.1 starts from the basic identity

tr R_{0}(f) = tr R′_{0}(f′)

valid for matching f and f′; see the Corollary to the proof of Theorem 1.1 in the preceding section. Equivalently, and more suggestively,

(2.2)  ∑_{π∈A_{0}(G)} tr π(f) = ∑_{π′∈A′_{0}(G)} tr π′(f′).

To extract the desired bijection between V_{π′} and π from the identity (2.2) we need to refine this identity to one of the following type:

Fix any finite set S of places of F containing S_{D} and the archimedean places, and for each v ∉ S, fix a given representation π_{v}^{0} which is unramified; then

(2.3)  ∑_{V_{π}} ∏_{v∈S} tr π_{v}(f_{v}) = ∑_{V_{π′}} ∏_{v∈S} tr π′_{v}(f′_{v}),

with f_{v} ∼ f′_{v}, and the sums taken over all V_{π} in A_{0}(G) (resp. V_{π′} in A_{0}(G)) such that π_{v} (resp. π′_{v}) is isomorphic to π_{v}^{0} for all v ∉ S. Note that by strong multiplicity one for GL(2), the left-hand side of (2.3) contains at most one term.

Assuming the truth of (2.3), let us explain the slick fashion in which Theorem 2.1 can be proved. Suppose first that no V_{π} in A_{0}(G) corresponds to a given V_{π′} in A′_{0}(G′). Then (2.3) (with S the set of places outside of which D and π′_{v} are p-adic and unramified, and π_{v}^{0} = π′_{v} for v ∉ S) implies that the left-hand side of (2.3) is zero. Thus we also have

(2.4)  ∑ tr π′_{S}(f′_{S}) = 0,

where

π′_{S} = ⊗_{v∈S}π′_{v}, f′_{S} = ∏_{v∈S} f′_{v}

is arbitrary in C_{c}^{∞}(Z_{S}(𝔸)\G′_{S}(𝔸)), and the sum is over all elements of A′_{0}(G′) (like, for example, V_{π′}) such that (π)_{v} ≅ π_{v}^{0} for v ∉ S. But (2.4) contradicts a well-known result of local harmonic analysis known as “linear independence of characters” (see Lemma 16.1.1 of [JL] or Lemma 5.11 of [Ro1]).

Similarly, we can begin to characterize the image of this correspondence V_{π′} → π. Indeed, suppose π = ⊗π_{v} is in the image of this correspondence, but π_{v} is not square-integrable for some v_{0} in S_{D}. Then it follows

tr π_{v_{0}}(f_{v_{0}}) = 0,

since π_{v_{0}} would then be induced, and f_{v_{o}} ∼ f′_{v_{0}} implies that f_{v_{0}} has vanishing hyperbolic integrals. Thus the left-hand side of (2.3) would vanish, leading to the same contradiction as above.

To complete the characterization of the image, we need to recall some facts about the square-integrable representations of G_{v} and G′_{v}. In particular, let ⟨ , ⟩_{e} denote the inner product for the space of class functions on the set of regular elliptic elements of Z_{v}\G_{v} or Z_{v}\G′_{v} defined by

⟨f_{1}, f_{2}⟩_{e} = ∑_{T_{v}} \frac{1}{2} ∫_{Z_{v}\T_{v}^{reg}} δ(t)f_{1}(t)\overline{f_{2}(t)} dt,

the sum extending over the conjugacy classes of compact tori of G_{v} (conveniently confused with those of G′_{v}). For G′_{v}, the Peter–Weyl theorem and the Weyl integration formula imply that the characters of the irreducible representations of Z_{v}\G′_{v} comprise a complete orthonormal set with respect to ⟨ , ⟩_{e}. On the other hand, for G_{v} the so-called “orthogonality relations for square integrable representations” imply that the characters of the square-integrable irreducible representations of Z_{v}\G_{v} comprise at least an orthonormal set with respect to ⟨ , ⟩_{e} Thus, given any square-integrable representation π_{v} on G_{v}, v ∈ S_{D}, we can determine an irreducible π′′_{v} on G′_{v} by the condition

⟨χ_{π_{v}}, χ_{π′′_{v}}⟩_{e} = a_{π_{v}} ≠ 0;

we also fix f′_{v} in C_{c}^{∞}(Z_{v}\G′_{v}) such that

tr π′_{v}(f′_{v}) = \{ \begin{matrix}1 & if π′_{v} ≅ π′′_{v}; \\ 0 & otherwise. \end{matrix}

Namely, we take f′_{v} = \overline{χ}_{π′′_{v}}.

So suppose now that π in A_{0}(G) is such that π_{v} is square-integrable for each v ∈ S_{D}. Then the trace formula identity (2.3) implies that

(2.5)  ∏_{v∈S} a_{π_{v}} = ∑_{V_{π′}} 1,

with the sum taken over V_{π′} in A′_{0}(G) such that π′_{v} ≃ π′′_{v} for v in S, and π′_{v} ≃ π_{v} for v ∉ S. Indeed, for v ∈ S,

tr π_{v}(f_{v}) = ∫_{Z_{v}\G_{v}} f(g)χ_{π_{v}}(g) dg

= ∑_{\{T_{v}\}} \frac{1}{2} ∫_{Z_{v}\T_{v}^{reg}} δ(t)χ_{π_{v}}(t)Φ_{t}(f) dt

= ∑ \frac{1}{2} ∫ δ(t)χ_{π_{v}}(t)Φ_{t}(f′_{v}) dt

= ⟨χ_{π_{v}}, χ_{π′′_{v}}⟩_{e} = a_{π_{v}},

with

Φ_{t}(f′_{v}) = ∫_{T_{v}\G_{v}} f_{v}(g^{−1}tg) dg = ∫_{T_{v}\G′_{v}} f′_{v}(g^{−1}tg) dg = Φ_{t}(f′_{v}).

Claim. The sum on the right side of (2.5) has exactly one term in it.

Indeed, if there were none, the left-hand side of (2.5) would be zero (which is impossible by our assumption in a_{π_{v}}, whereas as if there were two or more terms, we would have

∏ |a_{π_{v}}| ≥ 2

(again impossible, since each |a_{π_{v}}| = |⟨χ_{π_{v}}, χ_{π′′_{v}}⟩_{e}| ≤ ∥χ_{π_{v}}∥_{e}∥χ_{π′′_{v}}∥_{e} ≤ 1). Thus we conclude there must be exactly one V_{π′} in A′_{0}(G′) corresponding to π in A_{0}(G), and the theorem is finally proved.

2. Embedding of Local Discrete Series in Cusp Forms.

Theorem 2.1. Let V be any finite number of finite places of F, and π_{v} a discrete series representation of Z_{v}\G_{v} for each v in V. Then there exists a cuspidal automorphic representation π of G(𝔸) such that π_{v} = (π)_{v} for each v in V.

Proof. We can assume that for at least one place v_{1} ∈ V, π_{v_{1}} is actually supercuspidal (by adding such a pair (v_{1}, π_{v_{1}}) into the collection {v, π_{v}} if necessary). Before applying the simple trace formula of Deligne-Kazhdan to this situation, we need to recall two important facts about local harmonic analysis on G_{v}.

For G_{v} non-archimedean, a function f_{v} in C_{c}^{∞}(Z_{v}\G_{v}) is called a pseudomatrix coefficient for the discrete series representation π_{v} of Z_{v}\G_{v} if tr π_{v}(f_{v}) = 1, but tr τ_{v}(f_{v}) = 0 for any tempered irreducible admissible representation τ_{v} not equivalent to π_{v}. That such pseudo-matrix coefficients exist in the generality of a general reductive G is a theorem of Bernstein, Deligne and Kazhdan (see [BDK],

[Clo] [BDKV]). Moreover, one knows that for such functions f_{v} and γ (regular) elliptic in G_{v},

Φ_{γ}(f_{v}) = ∫_{T_{v}\G_{v}} f_{v}(x^{−1}γx) dx = c\overline{χ_{π_{v}}}(γ)

for a non-zero constant c. In particular, the orbital integral Φ_{γ}(f_{v}) is not identically zero (since χ_{π_{v}} ≢ 0 on the regular elliptic elements; see [Clo]).

Remark. Suppose π_{v} is supercuspidal, and we denote one of its matrix coefficients suitably normalized by f_{v}(g). Then f_{v} belongs to C_{c}^{∞}(Z_{v}\G_{v}),

tr π_{v}(f_{v}) = 1,  and
tr τ_{v}(f_{v}) = 0

for any irreducible admissible τ_{v} not equivalent to π_{v}. Thus the terminology pseudomatrix coefficient is indeed apt.

Now given V and {π_{v}} as in the hypothesis of Theorem 2.1, fix

f = ∏_{v} f_{v} in C_{c}^{∞}(Z(𝔸)\G(𝔸))

such that (a) for each v ∈ V, f_{v} is a pseudo-matrix coefficient; and (b) for a fixed finite place v_{2} outside V, f_{v_{2}} is supported on the regular elliptic set of G_{v_{2}}. Then by the simple trace formula of Deligne-Kazhdan, for such f,

(2.2)  ∑_{π} tr π(f) = ∑_{γ} m_{γ}Φ_{γ}(f),

where the sum on the left is over the cuspidal automorphic π on Z(𝔸)\G(𝔸) such that (π)_{v} ≅ π_{v} for each v ∈ V, and the right-hand side is summed over the regular elliptic conjugacy classes in Z(F)\G(F). Thus to prove Theorem 2.1, it suffices to prove that this right-hand side (and hence the left-hand side) does not vanish for a particular special choice of such f.

Equivalently, write

Φ_{γ}(f) = ∏_{v} Φ_{γ}(f_{v}),

a product of local orbital integrals. It will suffice then to choose f such that Φ_{γ}(f) ≠ 0 for a particular elliptic regular γ_{0}, but is zero for all other γ.

To this end, first choose γ_{0} and f_{v_{2}} (still assumed to have support in the regular elliptic set) such that

Φ_{γ_{0}}(f_{v}) ≠ 0 for each v in V ∪ {v_{2}}.

(Recall that for each pseudo-matrix coefficient, Φ_{γ}(f_{v}) = c\overline{χ_{π_{v}}(γ)} is not identically zero on the elliptic regular set of G_{v} ….) This particular choice of γ_{0} will intersect the maximal compact K_{v} of G_{v} for almost all finite places outside V ∪ {v_{2}}. Let us denote this (infinite) set of places by V^{∗}, and for each v in V^{∗} set f_{v} equal to the unit element of H_{v} = H(G_{v}, K_{v}). Finally, choose f_{v} at the remaining finite places (outside V ∪ {v_{2}} ∪ V^{∗}) to be such that Φ_{γ_{0}}(f_{v}) ≠ 0.

With this refined choice of f = ∏ f_{v} (still arbitrary at the infinite places!) it is clear that

(2.3)  Φ_{γ}(f) ≠ 0

for some elliptic regular γ only if the coefficients of the characteristic polynomial of γ are v-integral for each v in V^{∗}, i.e. are rational with uniformly bounded denominator. But such rational numbers lie in a lattice of F. Thus we can choose f_{v} at the archimedean places to have support so small near γ_{0} as to intersect the conjugacy class of no other γ satisfying (2.3). With this choice of f = ∏ f_{v}, Φ_{γ}(f) ≠ 0 if and only if γ = γ_{0}, and the proof is complete. □

Concluding Remark. Crucial to our argument was the fact that the pseudomatrix coefficients f_{v}, v ∈ V, eliminated from (2.2) all cuspidal τ = ⊗τ_{v} with τ_{v} not equivalent to π_{v} for v in V. This worked in our case since the only nontempered τ_{v} which might satisfy tr τ_{v}(f_{v}) ≠ 0 is the trivial representation (when π_{v} is the Steinberg representation), and such a representation can never occur as a local component of a cusp form on GL(2). In general, an additional argument is necessary, involving some kind of limit-multiplicity argument (due to de George-Wallach); see [Ro1] or Appendix 3 of [BDKV].

(New) References

[BDK] Bernstein, J., Deligne, P., and Kazhdan, D, Trace Paley–Weiner theorem for reductive p-adic groups, J. Anal. Math. 47 (1986), 180–192.

[Clo] Clozel, L., Invariant harmonic analysis on the Schwartz space of a reductive p-adic group, in Harmonic Analysis on Reductive Groups, edited by W. Barker and P. Sally, Progress in Mathematics, vol. 101, Birkhäuser, Boston, 1991, pp. 101–121.

[Kot] Kottwitz, R., Tamagawa Numbers, Ann. of Math. 127 (1988), 629–646.

[Lai] Lai, K. F., Tamagawa Numbers of Reductive Algebraic Groups, Compositio Math., vol. 41, Kluwer Acad. Publ., Dordrecht, 1980, pp. 153–188.

[La2] Base Change for GL(2), Ann. of Math. Stud., vol. 96, Princeton University Press, Princeton, NJ, 1980.

[La3] , The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, RI, 1966, pp. 143–148.

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

The transition from Arthur’s first (“coarse”) form of the trace formula to a more explicit (“fine”) expansion results from formal operations on collections of functions called (G, M)-families, operations which pervade the “fine 𝔬-expansion” as well as the “fine χ-expansion”.

The purpose of this lecture is to describe some basic examples and properties of (G, M)-families, and to explain the pivotal role they play, especially in making explicit the spectral ramified terms J_{χ}(f).

1. Definitions and Basic Examples. Fix a Levi subgroup M of G, and let P(M) denote the set of not necessarily standard parabolic subgroups of G for which M is the Levi component.

Definition 1.1.

Suppose that for each P in P(M), c_{P} (Λ) is a smooth function on i𝔞_{M}^{∗}. Then the collection

{c_{P} (Λ): P ∈ P(M)}

is called a (G, M)-family if the following condition holds: if P and P′ are adjacent groups in P(M), and Λ belongs to the hyperplane spanned by the common wall of the chambers of P and P′, then

(*)  c_{P} (Λ) = c_{P′}(Λ).

Remark 1.2. The compatibility condition (*) here is equivalent to the property that whenever P and P′ are elements of P(M) contained in a given parabolic subgroup Q, and Λ belongs to i𝔞_{Q}^{∗}, then

c_{P} (Λ) = c_{P′}(Λ).

In particular, let

F(M)

denote the parabolic subgroups of G whose Levi components contain M; then for each Q in F(M) one may define a smooth function c_{Q}(λ) on i𝔞_{Q}^{∗} ⊂ i𝔞_{P}^{∗} through the formula

c_{Q}(Λ) = c_{P} (Λ).

for any parabolic P in P(M) contained in Q.

LECTURE VII. (G, M)-FAMILIES AND THE SPECTRAL J_{χ}(f) 63

Figure 1

Example 1.3. Let {X_{P} : P ∈ P(M)} be a collection of points in 𝔞_{M} with the property that for each pair (P, P′) of adjacent groups in P(M), X_{P} − X_{P′} is perpendicular to the hyperplane spanned by the common wall of the chambers of P and P′; pictorially: Then the resulting family of functions

c_{P} (Λ) = e^{Λ(X_{P})}

on i𝔞_{M}^{∗} is a (G, M)-family. Indeed for Λ on such a hyperplane,

\frac{c_{P} (Λ)}{c_{P′}(Λ)} = e^{Λ(X_{P} −X_{P′})} = 1.

Remark 1.4. The similarity here with Figure 2 of Lecture IV is not coincidental. Indeed, for any fixed x in G(𝔸), the points

{−H_{P} (x): P ∈ P(M)}

comprise an “orthogonal family” of points as depicted in the figure above; this is proved in Lemma 3.6 of [A8], where such “A_{M}-orthogonal” families of points were first introduced. (N.B. A collection of points {X_{P} } in 𝔞_{M} comprises an A_{M}- orthogonal family if an only if X_{P} − X_{P′} is a multiple of the coroot associated to the unique root in Δ_{P} ∩ −Δ_{P′}).)

Remark 1.5. If L is any Levi subgroup of G containing M, the more general notion of an (L, M)-family is defined exactly as above: for each P in P^{L}(M) (the parabolic subgroups of L with Levi component M), c_{P} (λ) is smooth on i𝔞_{M}^{∗}, and if P′ in P^{L}(M) is adjacent to P, then c_{P} and c_{P′} agree on the wall between the corresponding adjacent chambers.

For the next Lemma, we need to recall a function θ_{P} (Λ) defined on i𝔞_{M}^{∗} by

θ_{P} (Λ) = (a_{P})^{−1} ∏_{α∈Δ_{P}} Λ(α^{∨}),

where a_{P} is the covolume of a certain coroot lattice. (Again, there is the more general notion of θ_{P}^{Q} for any parabolic Q ⊃ P . . ..)

Lemma 1.6. For any (G, M)-family {c_{P} (Λ): P ∈ P(M)}, the function

c_{M}(Λ) = ∑_{P ∈P(M)} c_{P} (Λ)θ_{P} (Λ)^{−1},

initially defined away from the hyperplanes Λ(α^{∨}) = 0, extends to a smooth function on i𝔞_{M}^{∗}.

Proof. The only possible singularities along Λ(α^{∨}) = 0 occur in terms corresponding to those P for which α or −α is a simple root. But such groups occur in pairs of adjacent P, P′, where c_{P} and c_{P′} agree. I.e., the contribution of these P ’s to c_{M}(Λ) is

\frac{c_{P} (Λ)}{θ_{P} (Λ)} + \frac{c_{P′}(Λ)}{θ_{P′}(Λ)},

where the numerators are equal, and the denominators appear as negatives of one another; thus the singularities cancel. □

Remark. The functions c_{M}(Λ) arise naturally as the weight functions of the weighted orbital integrals described in Lecture IV. Indeed, consider the (G, M)- family

{c_{P} (Λ) = e^{Λ(X_{P})}; X_{P} = −H_{P} (x)}

recalled above. In [A8] it was shown that the function

c_{M}(Λ) = ∑_{P ∈P(M)} e^{Λ(X_{P})}θ_{P} (Λ)^{−1}

equals the Fourier transform of the characteristic function of the convex hull of the points X_{P} (pictured in Figure 1 above). Thus it follows (without appealing to the more general Lemma 1.6 above) that c_{M}(Λ) is a smooth function on i𝔞_{M}^{∗} (being the Fourier transform of a compactly supported function). In particular, c_{M}(0) is defined, and equal to the volume of this convex hull, namely the weight function v_{M}(x).

It turns out that (G, M)-families pervade even more extensively the spectral terms J_{χ}(f). Here, as we shall explain below, (G, M) families arise which are products of families of the above (geometric) type, with (G, M)-families defined in terms of intertwining operators.

2. Motivation and Examples Arising from the Spectral Distributions J_{χ}^{T} (f). Recall (from Lecture III) that

J_{χ}^{T} (f) = ∫_{Z(𝔸)G(F)\G(𝔸)} Λ_{2}^{T} K_{χ}(x, x) dx,

with

K_{χ}(x, y) = ∑_{P} \frac{1}{n(A_{P})} ∑_{σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} ∑_{ϕ} E(x, ρ(σ, λ)(f)ϕ, λ)\overline{E(y, ϕ, λ)} dλ.

Here P = M_{P} N_{P} runs over the “associated” parabolics of G, ϕ runs thru a K-finite basis for ρ(σ, λ), and σ runs through (classes of) irreducible unitary representations of M_{P} (𝔸) such that functions in Ind_{P (𝔸)}^{G(𝔸)} σ belong to L^{2}(P)_{χ}.

Formally then,

J_{χ}^{T} (f) = ∑_{P} ∑_{σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f, P) dλ,

where

Ψ_{σ}^{T} (λ, f, P) = \frac{1}{n(A_{P})} tr(Ω_{χ,σ}^{T}(P, λ)ρ(σ, λ)(f)),

and

(Ω_{χ,σ}^{T}(P, λ)ϕ′, ϕ) = ∫_{Z(𝔸)G(F)\G(𝔸)} Λ^{T} E(x, ϕ′, λ)\overline{Λ^{T} E(x, ϕ, λ)} dx.

But according to the inner product formula for truncated Eisenstein series, this operator Ω_{χ,σ}^{T}(P, λ) equals the value at λ′ = λ of

(2.1)  ∑_{P_{1}} ∑_{\begin{matrix}t,t′ in \\ W (𝔞_{P} ,𝔞_{P_{1}} ) \end{matrix}} M_{P_{1}|P} (t, λ)^{−1}M_{P_{1}|P} (t′, λ′)\frac{e^{(t′λ′−tλ)(T)}}{θ_{P_{1}}(t′λ′ − tλ)}.

Here M_{P_{1}|P} (t, λ) is the intertwining operator defined on Ind(σ, λ) by

∫_{N_{P_{1}} (𝔸)∩w_{t}N_{P} (𝔸)w_{t}^{−1}\N_{P_{1}} (𝔸)} ϕ(w_{t}^{−1}nx) dx

(initially for λ with large real part, then for any λ by analytic continuation). Denote by ω_{χ,σ}^{T}(P, λ) the value of the operator defined in (2.1) at λ′ = λ. Actually, for P ∉ P_{χ}, the operators Ω_{χ,σ}^{T}(P, λ) and ω_{χ,σ}^{T}(P, λ) are only asymptotically equal (for large T); this is the thrust of [A9].

Claim.

(2.2)  tr(ω_{χ,σ}^{T}(P, λ)ρ(σ, λ)(f))

may be evaluated by setting λ′ = λ in the sum over s ∈ W (𝔞_{P} , 𝔞_{P}) of

tr( ∑_{Q∈P(M)} c_{Q}(Λ)d_{Q}(Λ)θ_{Q}(Λ)^{−1}ρ(σ, λ)(f)),

where Λ = Λ_{s} = sλ′ − λ, and c_{Q} and d_{Q} are two (G, M) families to be described below.

To see this, rewrite the expression (2.1) as

∑_{\begin{matrix}s∈W_{P} = \\ W (𝔞_{P} ,𝔞_{P}) \end{matrix}} ∑_{P_{1}} ∑_{t∈W (𝔞_{P} ,𝔞_{P_{1}} )} M_{P_{1}|P} (t, λ)^{−1}M_{P_{1}|P} (ts, λ′)\frac{e^{(t(sλ′−λ))(T)}}{θ_{P_{1}}(t(sλ′ − λ))}.

For any P_{1} and t ∈ W (𝔞_{P} , 𝔞_{P_{1}}), set Q = w_{t}^{−1}P_{1}w_{t} for any representative w_{t} of t in G(F). Then

(P_{1}, t) ↔ Q

is a bijection between pairs which occurs in the sum above and groups Q ∈ P(M). It can also be checked that θ_{P_{1}}(t(sλ′ − λ)) = θ_{Q}(sλ′ − λ); moreover,

M_{P_{1}|P} (t, λ)^{−1}M_{P_{1}|P} (ts, λ′)e^{(t(sλ′−λ))(T)} = M_{Q|P} (λ)^{−1}M_{Q|P} (s, λ′)e^{(sλ′−λ)Y_{Q}(T)},

where M_{Q|P} (λ) = M_{Q|P} (1, λ), and

Y_{Q}(T) = t^{−1}T

(or rather the projection of this point onto 𝔞_{M}). Thus the expression (2.1) indeed equals the value at λ′ = λ of the sum over s in W (𝔞_{P} , 𝔞_{P}) of

∑_{Q∈P(M)} c_{Q}(Λ)d_{Q}(Λ)θ_{Q}(Λ)^{−1}

with

(2.3)  c_{Q}(Λ) = e^{Λ(Y_{Q}(T))},
Λ = sλ′ − λ,

and

(2.4)  d_{Q}(Λ) = (M_{Q|P} (λ)^{−1}M_{Q|P} (s, λ′)).

Remark 2.5. Below we shall see that {c_{Q}(Λ)} and {d_{Q}(Λ)} both define (G, M)- families. Thus the product

{e(Λ) = c(Λ)d(A)}

is also a (G, M)-family, and the corresponding function

e_{M}(Λ) = ∑_{Q∈P(M)} c_{Q}(Λ)d_{Q}(Λ)θ_{Q}(Λ)^{−1}

is smooth on i𝔞_{M}^{∗}. In particular, the expression (2.2) may indeed be evaluated at λ′ = λ (which corresponds to evaluating e_{M}(Λ) at Λ = sλ − λ).

Remark 2.6. In (2.3) and (2.4), set λ′ = ζ + λ in i𝔞_{M}^{∗}, and s = the identity. Then the value of e_{M}(Λ) at λ′ = λ corresponds to the (operator) expression

val_{λ′=λ} ∑_{Q∈P(M)} e^{Λ(Y_{Q}(T))} \frac{M_{Q|P} (λ)^{−1}M_{Q|P} (λ′)}{θ_{Q}(ζ)}

= lim_{ζ→0} ∑_{P_{1}} ∑_{w∈W (𝔞_{P} ,𝔞_{P_{1}} )} e^{wζ(T)} \frac{M(w, λ)^{−1}M(w, λ + ζ)}{∏_{α∈Δ_{P_{1}}} (wζ)(α^{∨})}

This is precisely the operator we denoted by M_{P}^{T} (σ)_{χ} in Proposition 3.9 of Lecture IV. Thus the “weight function” M_{P}^{T} (σ)_{χ} appearing in the weighted character expression for the unramified spectral term J_{χ}^{T} (f) in Proposition 3.9 is indeed realizable as a special value of e_{M}(Λ) for an appropriate (G, M)-family e_{P}.

N.B. As we observed in the case of GL(2) (in §2 of Lecture IV), the expression for J_{χ}^{T} (f) simplifies considerably for unramified spectral data χ, since the sum over t, t′ in W (𝔞_{P} , 𝔞_{P_{1}}) in (2.1) need only be taken over t = t′, i.e., only the term corresponding to s = 1 really appears in the expression for (2.2), and we indeed have

J_{χ}^{T} (f) = ∑_{P ∈P_{χ}} ∑_{σ} \frac{1}{n(A_{P})} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} tr(M_{P}^{T} (σ)_{χ}ρ(σ, λ)) dλ

as asserted in Proposition 3.9 of Lecture IV.

Proposition 2.7. Both c_{Q}(Λ) and d_{Q}(Λ) define (G, M)-families.

Proof. We have to show that if Q and Q′ are adjacent in P(M), then

Y_{Q}(T) − Y_{Q′}(T)

is perpendicular to the hyperplane in i𝔞_{M}^{∗} spanned by the wall common to the chambers of Q and Q′. Recall that for any Q in P(M), Y_{Q}(T) = t^{−1}T if Q = w_{t}^{−1}P_{1}w_{t} for some fixed P_{1} ∈ P(M). But if Q′ is adjacent to Q, then Q′ = w_{t′}^{−1}P_{1}w_{t′} with t′ = s_{α}t, and s_{α} the simple reflection corresponding to an α ∈ Δ_{P_{0}}. Therefore Y_{Q}(T) − Y_{Q′}(T) = t^{−1}(T − s_{α}^{−1}T), which is a multiple of t^{−1}α^{∨} (see footnote^{1} below). But s^{−1}α^{∨} = β^{∨}, where β^{∨} is the unique root in Δ_{Q} ∩ (−Δ_{Q′}). Thus Y_{Q}(T) − Y_{Q′}(T) is perpendicular to the hyperplane between Q and Q′. □

As for d_{Q}(Λ), we recall the standard functional equation

M_{Q′|P} (s, λ) = M_{Q′|Q}(1, sλ)M_{Q′|P} (s, λ),

valid for s ∈ W (𝔞_{M}, 𝔞_{M}) and Q ∈ P(M) (see equation (1.2) of [A8]). Using it gives

d_{Q′}(Λ) = M_{Q′|P} (λ)^{−1}M_{Q′|P} (s, λ′)
= M_{Q|P} (λ)^{−1}M_{Q′|Q}(λ)^{−1}M_{Q′|Q}(sλ′)M_{Q|P} (s, λ′).

So it remains to show that for Λ belonging to the hyperplane common to Q′ and Q,

M_{Q′|Q}(λ)^{−1}M_{Q′|Q}(sλ′) = I.

For this, see §§1 and 2 of [A8].

1Formally, α^{∨} = 2α/(α, α), and s_{α}^{−1} = T − 2(T, α)/(α, α) α. Therefore, for any β, (β, T − s_{α}^{−1}(T)) = (β, T)−(β, T)+2(T, α)/(α, α) (β, α) = (T, α)(β, α^{∨}), i.e., T −s_{α}^{−1}T is indeed a multiple of α^{∨}.

3. A Splitting Formula for Products of (G, M)-families. In the last section, we explained how J_{χ}^{T} (f) is expressed in terms of integrals of special values of the functions e_{M}(Λ), where e_{P} = c_{P} (Λ)d_{P} (Λ) is the product of two (G, M)-families. The result below is crucial in completing the evaluation of these integrals.

Proposition 3.1 (see Lemma 6.3 of [A4]). In general,

(cd)_{M}(Λ) = ∑_{Q∈F(M)} c_{M}^{Q}(Λ)d′_{Q}(Λ),

where F(M) denotes the set of parabolics of G whose Levi component contains (but does not necessarily equal) M.

We will not prove this proposition, since the reader can find it in [A4]. But we shall at least define the terms c_{M}^{Q} and d′_{Q} appearing here, and explain their significance.

Given any (G, M)-family {c_{P} (Λ)}, and a parabolic Q containing some P in P(M), how first do we define the function c_{Q}?

Since i𝔞_{Q}^{∗} ⊂ i𝔞_{P}^{∗}, it is tempting to simply set

c_{Q}(Λ) = c_{P} (Λ_{Q})

with Λ_{Q} the projection of Λ onto i𝔞_{Q}^{∗}. But is this well-defined? If P′ is another element of P(M) contained in Q, we must check that

c_{P′}(Λ) = c_{P} (Λ)

as soon as λ ∈ i𝔞_{Q}^{∗}. But we have already remarked that the compatibility condition on the (G, M)-family {c_{P} } implies this is the case (Exercise!), and thus c_{Q} is welldefined on i𝔞_{Q}^{∗}.

Now for any pair of parabolics Q ⊂ R, define

θ̂_{Q}^{R}(Λ) = (â_{Q}^{R})^{−1} ∏_{w̃∈Δ̂_{Q}^{R}} Λ(w̃^{∨}),

with w̃^{∨} in 𝔞_{Q}^{R} defined by

α(w̃^{∨}) = w̃(α^{∨}), α ∈ Δ_{Q}^{R}

(and the constant â_{Q}^{R} the covolume of a certain lattice …). Then the functions

c′_{Q}(Λ), Q ⊃ P

are defined (initially on the complement of a finite set of hyperplanes in i𝔞_{P}^{∗}) by

c′_{Q}(Λ) = ∑_{{R: R⊃Q}}(−1)^{dim(A_{Q}|A_{R})}θ̂_{Q}^{R}(Λ)^{−1}c_{R}(Λ)θ_{R}(Λ)^{−1}.

In Lemma 6.1 of [A4], it is shown that these functions extend to smooth functions on all of i𝔞_{P}^{∗}.

Now that the functions d′_{Q} in Proposition 3.1 are defined, there remains the simpler task of defining the c_{M}^{Q}(Λ); these are just the c_{M} functions (1.5) derived from the following (L, M)-families of functions {c_{R}^{Q}(Λ)}.

Fix Q arbitrary in F(M), and let L = L_{Q} denote its Levi component. By P^{L}(M) we denote the set of parabolic subgroups of L whose Levi component is M, and for each R ∈ P^{L}(M), we set Q(R) equal to the unique group in P(M) which is contained in Q and such that Q(R)∩L = R. Then the resulting family of functions

{c_{R}^{Q}(Λ) = c_{Q(R)}(Λ): R ∈ P^{L}(M)}

comprises an (L, M)-family, and the functions c_{M}^{Q}(Λ) appearing in Proposition 3.1 are the corresponding c_{M}-functions

c_{M}^{Q}(Λ) = ∑_{R∈P^{L}(M)} c_{R}^{Q}(Λ)θ_{R}^{Q}(Λ)^{−1}.

4. GL(2) Revisited. Although we already discussed (in Lecture IV) the evaluation of J_{χ}^{T} (f) in this case, we wish to return to this calculation now from the more general standpoint of Arthur’s theory.

According to the discussion in Section 2 above, J_{χ}^{T} (f) is given by the polynomial

(4.1)  ∑_{P} ∫_{i𝔞_{P}^{∗} \i𝔞_{G}^{∗}} ∑_{σ} ∑_{s∈W (𝔞_{P})}{λ = λ′^{val} tr(e_{M}(Λ)ρ(σ, λ)(f))} dλ,

where e(Λ) is the (G, M)-family given by the product of the family of functions (2.3) and (2.4). When P = B (the Borel subgroup), we have

λ = (\frac{it}{2}, \frac{−it}{2})

and

λ′ = (\frac{it′}{2} , \frac{−it′}{2} ),

with t′ = t + ζ and Λ = sλ′ − λ.

For χ = {(M, μ)}, the contribution from the term s = 1 reduces to (4.2)

∫_{ℝ} tr (lim_{ζ→0}\{\frac{e^{iζ(T_{1}−T_{2})/2}}{iζ} + \frac{e^{−iζ(T_{1}−T_{2})/2}M(w, it)^{−1}M(w, i(t + ζ))}{−iζ}\}ρ(μ, it)(f)) dt

Indeed, the two elements of P(M) are just B and B̄ (the subgroup wBw^{−1} opposite to B), and so

e_{M}(Λ) = e_{B}(Λ)θ_{B}(Λ)^{−1} + eB̄(Λ)θ_{B}(Λ)^{−1}

= \frac{c_{B}(Λ)d_{B}(Λ)}{iζ} + \frac{cB̄(Λ)dB̄(Λ)}{−iζ}.

But (2.3) clearly implies that

c_{B}(Λ) = e^{Λ(T)} = e^{iζ(T_{1}−T_{2})/2}

and

cB̄(Λ) = e^{−Λ(T)} = e^{−iζ(T_{1}−T_{2})/2}.

Also M_{B|B}(1, λ) = I, so d_{B}(Λ) = I, and it remains only to compute dB̄(Λ). For this, let T_{w} denote the linear transformation ϕ(x) ⟶ ϕ(wx) taking IndB̄ σ^{w} | |^{wλ} to Ind_{B} σ | |^{λ}. Then it is straightforward to check that

dB̄(Λ) = M_{B̄|B}(λ)^{−1}M_{B̄|B}(1, λ′)
= (T_{w}M(w, λ))^{−1}(T_{w}M(w, λ′))
= M(w, it)^{−1}M(w, i(t + t′)),

and thus (4.2) holds. To continue, there is no need to appeal to the fact that the expression in brackets in (4.2) comes from the (G, M)-family {e_{P} }. Instead, one can just compute the limit in question to be

(T_{1} − T_{2}) + M(−it)M′(it),

and hence the contribution to J_{χ}^{T} (f) (from s = 1) to be

(T_{1} − T_{2}) ∫_{=∞}^{∞} tr(ρ(μ, it)(f) dt + ∫_{=∞}^{∞} tr(M(−it)M′(it)ρ(μ, it)(f) dt.

This of course agrees with formula (2.5) of Lecture IV, and for χ unramified, it accounts for all of J_{χ}^{T} (f).

Now let us concentrate on the more interesting contribution to (4.1), namely the one corresponding to s = w. In this case, Λ = wλ′ − λ, and this equals (−it, it) when λ′ = λ. So as above, we compute

c_{B}(Λ) = e^{Λ(T)} = e^{−2it(T_{1}−T_{2})}
cB̄(Λ) = e^{−Λ(T)} = e^{2it(T_{1}−T_{2})}
d_{B}(Λ) = M(w, λ′)

and

dB̄(Λ) = M(w, λ)^{−1}.

In other words, for s = w

{λ′ = λ^{val} tr(e_{M}(Λ)ρ(σ, λ)(f)}

= tr(M(w, −it)(ρ(μ, it)(f))\frac{e^{2it(T_{1}−T_{2})}}{2it} − tr(M(w, it)(ρ(μ, it)(f))\frac{e^{−2it(T_{1}−T_{2})}}{2it} .

This is exactly the expression (6.28) in [GJ]. But instead of evaluating its integral over ℝ using the simple manipulations of [GJ], let us instead appeal to the splitting formula for e_{M}(Λ).

Lemma. For s = w

e_{M}(Λ) = (cd)_{M} = d_{B}(0)\frac{e^{Λ(T)} − e^{−Λ(T)}}{−2it}

+ \frac{d_{B}(Λ) − d_{B}(0)}{−2it} · e^{Λ(T)}

+ \frac{dB̄(Λ) − d_{B}(0)}{2it} · e^{−Λ(T)}

with e^{±Λ(T)} = e^{∓2it(T_{1}−T_{2})}.

Proof. According to Proposition 3.1, we need to compute

c_{M}^{Q}(Λ)d′_{Q}(Λ)

for each Q ∈ F(M). So first fix Q = G. By definition, d_{G}(Λ) = d_{B}(0) = dB̄(0), and then d′_{G}(Λ) = d_{G}(Λ). On the other hand, we compute (from the definition of c_{M}^{Q}) that c_{M}^{G}(Λ) = c_{M}(Λ). Thus the leading term of (cd)M(Λ) is indeed

d_{B}(0)\frac{e^{Λ(T)} − e^{−Λ(T)}}{−2it} ,

as claimed.

Now fix Q = B. Then

d′_{B}(0) = \frac{d_{B}(Λ) − d_{B}(0)}{−2it},

while c_{M}^{B}(Λ) = c_{B}(Λ). Indeed, in this case L = M and P^{L}(M) = M! Similarly

cB̄ ; M(Λ)d′ ; B̄(Λ) = \frac{dB̄(Λ) − dB̄(0)}{−2it} · cB̄(Λ),

and we’re done. □

Proposition. For T large, the contribution to J_{χ}^{T} (f) from (the Borel subgroup B and) s = w is just

tr(d_{B}(0)ρ(0, μ)(f)) = tr(M(0, μ)ρ(0, μ)(f));

in particular, only the leading term in (cd)_{M}(Λ) contributes to J_{χ}^{T} (f).

Proof.

Let us suppose for the moment that J_{χ}^{T} (f) can be calculated by taking a compactly supported Schwartz function B(t) with B(0) = 1, and then computing

J_{χ}^{T} (f) = lim_{ε→0} lim_{α(T)→∞} ∫_{−∞}^{∞} tr(ω_{χ}^{T} (t)ρ(μ, it)(f))B(εt) dt.

By the last lemma, the contribution from s = w (and P = B) to

tr(ω_{χ}^{T} (t)ρ(μ, it)(f))

is

\{\frac{e^{2it(T_{1}−T_{2})} − e^{−2it(T_{1}−T_{2})}}{2it} \} tr(d_{B}(0)ρ(μ, it)(f))

+ e^{−2it(T_{1}−T_{2})} \frac{tr((d_{B}(Λ) − d_{B}(0))ρ(μ, it)(f))}{−2it}

+ e^{2it(T_{1}−T_{2})} \frac{tr((dB̄(Λ) − d_{B}(0))ρ(μ, it)(f))}{−2it}.

But for any real T,

\frac{e^{iyT}}{iy} + \frac{e^{−iyT}}{−iy} = ∫_{−T}^{T} e^{iyx} dx
= ∫_{∞}^{∞} χ_{T} (x)e^{−iyx} dx,

with χ_{T} the characteristic function of [−T, T]. Thus we can write the contribution of the leading term as

lim_{ε→0} lim_{α(T)→∞} {∫_{−∞}^{∞} χ_{α(T)}(x) ∫_{−∞}^{∞} e^{2itx} tr(d_{B}(0)ρ(μ, it)(f))B(εt) dt dx}

= lim_{ε→0} lim_{α(T)→∞} {∫_{−∞}^{∞} χ_{α(T)}(x)F̂_{ε}(x) dx}

= lim_{ε→0} {∫_{−∞}^{∞} F̂_{ε}(x) dx} = lim_{ε→0} F_{ε}(0),

where

F_{ε}(t) = tr(d_{B}(0)ρ(μ, it)(f))B(εt)
= tr(M(0, μ)ρ(μ, 0)(f))  at t = 0,

since B(0) = 1. On the other hand, for the second (or third) term, the contribution is

lim_{ε→0} lim_{α(T)→∞} ∫_{−∞}^{∞} e^{∓2it(T_{1}−T_{2})}F_{ε}(t) dt

where for each ε,

F_{ε}(t) = tr \{\frac{d_{B}(t) − d_{B}(0)ρ(μ, it)(f)}{±2it}\} B(εt)

is a smooth Schwartz function on ℝ, whose Fourier transform vanishes at infinity. Thus the second and third terms contribute nothing to J_{χ}^{T} (f). □

Concluding Remarks. (1) In treating the GL(2) calculation of J_{χ}^{T} (f) from the point of view of Arthur’s general theory, we have made a big mountain out of a small mole hill. However, already for GL(3), there are compelling reasons for appealing to the general theory—not just for using the splitting formula for products of (G, M)-families, but also for exploiting the compactly supported function B in the calculation of J_{χ}(f). Indeed, without using this function B, there is the need to directly establish the integrability of a function like tr(M(w, λ)ρ(σ, λ)(f)) (in order to justify applying a Riemann-Lebesgue Lemma as T → ∞). In the case of GL(2), this can be checked directly, but already for GL(3) the required estimates on the growth of M(w, λ) are non-trivial (see Section 3 of [Ja1]), and in general they are simply not available.

(2) It remains to prove that the above computation (with the insertion of the “smoothing function” B(t)) is justified. This is precisely the goal of Arthur’s paper [A7] and—we hope—part of the subject matter of Lecture IX.

(New) Reference

[Ja1] Jacquet, H., The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension, Israel Journal of Math. 89, 1–59.

Lecture VIII. Jacquet’s Relative Trace Formula

In deriving Arthur’s trace formula

(*)  ∑_{𝔬} J_{𝔬}^{T} (f) = ∑_{χ} J_{χ}^{T} (f),

we started by restricting the kernel functions

K_{f}(x, y) = ∑_{𝔬} K_{𝔬}(x, y) = ∑_{χ} K_{χ}(x, y)

to the diagonal subgroup.

Z(𝔸)G(F)\G(𝔸) × Z(𝔸)G(F)\G(𝔸).

Then after modifying these restricted kernels, we were able to integrate them and obtain the formula (*). The term “trace formula” seemed apt because—at least for cuspidal data χ = {(M, r)} with M = G, J_{χ}^{T} (f) actually represents the trace of R(f) restricted to L_{χ}^{2}.

The idea of Jacquet’s relative “trace formula” is to restrict the kernel functions K_{f}(x, y) to interesting subgroups of the diagonal, and then integrate them against possibly non-trivial automorphic forms on these subgroups. Although the resulting integrals no longer represent the “trace” of anything, there results from this new approach a dramatic array of interesting possible applications to automorphic forms (See [JLR] and [Ja1] for a general discussion, and further references.)

In this lecture we shall concentrate on a particular example (cf. [Ye]) which already shows the power of this method, and indicates how the further development of the method is inextricably linked to the general theory developed by Arthur. Even in our exposition of this simplest example, we are influenced by Jacquet’s more general ideas (see [Ja1]); moreover, the approach to the computations in §4 is adapted from joint work with him and Rogawski on similar computations for GL(3).

1. Description of a Base Change Theorem. Fix a quadratic extension E of the number field F, and let ω_{E/F} denote the corresponding quadratic character of F^{∗}\𝔸_{F}. Let us call a cuspidal automorphic representation π of GL_{2}(𝔸_{F}) stable if it is not of the form π(μ′) for any grossencharacter μ′ of E.

Theorem (cf. [Ye]). Every stable cuspidal π on GL_{2}(𝔸_{F}), with central character ω_{E/F}, has a base change lift to a cuspidal representation π_{E} of GL_{2}(𝔸_{E}); moreover, the resulting map

π ⟶ π_{E}

is a bijection onto the cuspidal representations Π of GL_{2}(𝔸_{E}) which (have trivial central character and ) are distinguished with respect to GL_{2}(F), i.e., the “period”

(1.2)  ∫_{Z(𝔸) GL_{2}(F)\ GL_{2}(𝔸_{F})} ϕ′(h) dh

is non-zero for some ϕ′ in the space of Π.

Remarks. (i) In [La2], Langlands established the existence of a base change lift for GL_{2} for an arbitrary cyclic extension E of F, and in[AC] these results were dramatically generalized to GL_{n}. Both these works characterize the image of base change in terms of the Galois invariance of the representations “upstairs”; this is not surprising, since the proofs proceed from a comparison of the (“invariant”) trace formula over F with the (Galois) “twisted” (“invariant”) trace formula over E.

(ii) In [HLR] it is shown that an automorphic cuspidal representation Π of GL_{2}(E) (trivial on its center) is the base change lift of an automorphic cuspidal representation π of GL_{2}(𝔸_{F}) (with central character ω_{E/F}) if and only if Π is distinguished. The argument there combines the base change result of [La2] with properties of the so-called Asai L-function L(s, Π, Asai) (which has a pole at s = 1 if and only if π is distinguished …). (By contrast, the relative trace formula proof of Theorem 1.1 (see below) makes no appeal to [La2].) The main thrust of [HLR] was to study algebraic cycles on certain Shimura varieties attached to GL_{2} over E, whose Hasse–Weil zeta functions are computable in terms of

L(s, Π, Asai).

It was precisely the desire to extend these results to the context of quaternion algebras that led to the idea of the relative trace formula introduced in [JLai].

(iii) In [JLai], Jacquet and Lai used a comparison of relative trace formulas on GL_{2} /E and the multiplicative group G′ of a division quaternion algebra over E (the relative subgroups of the diagonal being the F -diagonal subgroups) to prove that Π′ on G′/E is G′(F) distinguished if and only if its Jacquet–Langlands correspondent Π on GL_{2} /E is GL_{2}(F) distinguished. This was applied in [Lai2] to extend the results of [HLR] to the case of certain compact Shimura varieties.

2. Outline of the Proof of Theorem 1.1. It is convenient to use the notation H for the group GL_{2} over F, and G for the group Res_{F}^{E} GL_{2} ( so that H(𝔸) = GL_{2}(𝔸_{F}) and G(𝔸) = GL_{2}(𝔸_{E})). By

f = ∏ f_{v}

we denote a test function in C_{c}^{∞}(H(𝔸), ω_{E/F}), where for almost every place v, f_{v} is the identity element of the local Hecke algebra H(H_{v}, K_{v}^{H}, ω_{v}); similarly,

f′ = ∏ f′_{v}

belongs to C_{c}^{∞}(Z(𝔸_{E}\G(𝔸_{F})), with f′_{v} almost everywhere the identity in

H(Z_{G_{v}}\G_{v}, K_{v}^{G}).

Corresponding to f there is the convolution operator

R(f) = ∫_{Z(𝔸)\H(𝔸)} f(h)R(h) dh

in the space L^{2}(H(F)\H(𝔸), ω); it is an integral operator with kernel

K_{f}(x, y) = ∑_{γ∈Z(F)\H(F)} f(x^{−1}γy).

Similarly f′ determines an integral operator R′(f′) in L^{2}(Z_{G}(𝔸)G(F)\G(𝔸)) with kernel

K_{f′}(g_{1}, g_{2}) = ∑_{γ∈Z_{G}(F)\G(F)} f′(g_{1}^{−1}γg_{2}).

The proof of Theorem 1.1 will come from comparing certain “relative traces” associated to these kernels, as we shall now explain.

Step I. Prove that for “matching” f and f′,

(2.1) ∫∫_{[N(F)\N(𝔸_{F})]^{2}} K_{f}(n_{1}, n_{2})ψ_{N}(n_{1}^{−1}n_{2}) dn_{1} dn_{2}

= ∫_{Z(𝔸)H(F)\H(𝔸)} ∫_{N′(F)\N′(𝔸_{F})} K_{f′}(n′, h)ψ′_{N′}(n′) dn′dh.

Here N (resp. N′) is the standard maximal unipotent of H (resp. G), and ψ_{N} (resp. ψ′_{N}) the corresponding character of N(F)\N(𝔸_{F}) (resp. N′(F)\N′(𝔸_{F})) determined by a fixed additive character ψ (of F \𝔸) (with ψ′ = ψ ∘ tr_{E/F}).

Remarks. (i) A special feature of this particular relative trace formula approach is the absolute convergence of the integrals on either side of the identity (2.1). On the left hand side, this is clear from the compactness of the domains of integration; for the right side, this follows from the fact that for sufficiently large T,

∫_{N′(F)\N′(𝔸)} Λ_{2}^{T} K_{f′}(n′, h)ψ′_{N′}(n′) dn′

equals (as opposed to just approximates)

∫ K_{f′}(n′, h)ψ′_{N′}(n′) dn′.

For a proof, see Proposition 2.1 of [Ja1].

(ii) At first sight, the equality in (2.1) seems surprising, since the double cosets

N\H/N and N′\G/H

which arise on each side do not match up. However, it seems to be a general feature of the relative trace formula that the “relevant” orbits—those which survive the relative integrations—do indeed match up. In the case at hand, it is the integration against ψ which allows for the matching up of relevant orbits. Once this is established, the assertion that (2.1) holds for matching f and f′ reduces quickly to a collection of purely local assertions: two local function f_{v} and f′_{v} are “matching” if certain local relative orbital integrals (indexed by identical sets) are equal.

(iii) It is also crucial to prove a “fundamental lemma” asserting that this local matching is compatible with the base change map BC ∗ between the Hecke algebras of K_{v} and G_{v}. This map is dual to the natural base change morphism

BC : (g, σ) ⟶ (g, g, σ)

from ^{L}H = GL_{2}(ℂ) × Gal(E/F) to ^{L}G = (GL_{2}(ℂ) × GL_{2}(ℂ)) ⋊ Gal(E/F). In particular, when f′_{v} belongs to the Hecke algebra of G_{v}, the function f_{v} in the Hecke algebra of H_{v} defined by

(f_{v})^{∨}(t) = (f′_{v})^{∨}(BC(t))

will be a matching function to f′_{v}. For proofs of these matching results, see [Ye]. For more general results, and discussion of “relative fundamental lemmas”, see [Ja2], [JaYe], [Mao], and [JLR].

Step II. Prove (again for these matching f and f′) that the relative traces of the continuous parts of the kernels K_{f} and K_{f′} cancel each other out.

To explain what this means, let us first write out the spectral expansion of K_{f} in crude form:

K_{f}(x, y) = K_{f, cont} + K_{f,sp} + K_{f,0},

where K_{f,cont} denotes the kernel of R(f) restricted to the continuous spectrum of L^{2}(H(F)\H(𝔸), ω_{E/F}), and K_{f,sp} the kernel for the discrete non-cuspidal spectrum. Similarly,

K_{f′}(g_{1}, g_{2}) = K_{f′,cont} + K_{f′,sp} + K_{f′,0}.

A simple but key observation is that the sp-part of the kernel contributes nothing to the relative trace. Indeed,

∫_{[N(F)\N(𝔸_{F})]^{2}} K_{f,sp}(n_{1}, n_{2})ψ(n_{1}^{−1}n_{2}) dn_{1} dn_{2}

= 0 = ∫_{N′(F)\N′(𝔸)} K_{f′,sp}(n′, h)ψ′_{N′}(n′) dn′, since these parts of the spectrum are non-generic.

Thus, if we can show that the continuous kernels contribute roughly equal relative traces, we might then conclude

(2.2)  ∫∫ K_{f,0}(n_{1}, n_{2})ψ_{N}(n_{1}^{−1}n_{2}) dn_{1} dn_{2} = ∫∫ K_{f′,0}(n′, h)ψ_{N′}(n′) dn′ dh

Henceforth, we denote the maximal compact of H_{v} by K_{v} and the maximal compact of G_{v} by K′_{v}.

Step III. Use the equality of the relative cuspidal traces on H and G to establish the asserted base change bijection between the cuspidal (stable) π on H and the cuspidal (distinguished) Π on G.

To see why this is relatively (!) straightforward, let us write out the explicit spectral expansions for K_{f,0} and K_{f′,0}, namely,

K_{f,0}(x, y) = ∑_{π ; cuspidal} ∑_{ϕ∈V_{π}} π(f)ϕ(x)\overline{ϕ(y)},

and

K_{f′,0}(g_{1}, g_{2}) = ∑_{Π cuspidal} ∑_{ϕ′∈V_{Π}} Π(f)ϕ(g_{1})\overline{ϕ′(g_{2})}.

If we assume f and f′ are right K^{S} = ∏_{v∉S} K_{v} (resp. K′^{S}) invariant, where S is a finite set of places (including the infinite ones), then these identities read

(2.3)  K_{f,0}(x, y) = ∑_{π^{K^{S}} ≠\{0\}}(f^{S})^{∨}(t(π^{S})) ∑_{\{ϕ_{π}\}} ∏_{v∈S} π_{v}(f_{v})ϕ(x)\overline{ϕ(y)}

and

(2.3′)  K_{f′,0}(g_{1}, g_{2}) = ∑_{Π^{^{K′}^{S}} ≠\{0\}}(f′^{S})^{∨}(t(Π^{S})) ∑_{\{ϕ′_{Π}\}} ∏_{v∈S} Π_{v}(f′_{v})ϕ′(g_{1})\overline{ϕ′(g_{2})}.

If we further assume that for v ∉ S (where both f′_{v} and f_{v} are spherical), f_{v} corresponds to f′_{v} via the natural base change map between their Hecke algebras, then,

(f^{S})^{∨}(t(π^{S})) = (f′^{S})^{∨}(BC(t(π^{S}))).

So plugging these expressions into the relative trace formula identity (2.2) yields the more explicit identity

(2.4)  ∑_{π} A(π, f_{S})(f′^{S})^{∨}(BC(t(π^{S})) = ∑_{Π} B(Π, f′_{S})(f′^{S})^{∨}(t(Π^{S}))

with

A(π, f_{S}) = ∑_{\{ϕ_{π}\}} W_{π_{S}(f_{S})ϕ}^{ψ}(e)\overline{W_{ϕ}^{ψ}(e)},
B(Π, f′_{S}) = ∑_{\{ϕ′_{Π}\}} W_{Π_{S}(f′_{S})ϕ′}^{ψ′}(e)\overline{D(ϕ′)},

W_{ϕ}^{ψ}(e) = ∫_{N(F)\N(𝔸)} ϕ(n)\overline{ψ_{N}(n)} dn,  (with a similar expression for W^{ψ′})

and

D(ϕ′) = ∫_{Z(𝔸)H(F)\H(𝔸)} ϕ′(h) dh.

Note that (2.4) may be viewed as a (absolutely convergent) linear combination of irreducible characters of the group (G)^{S} = ∏_{v∉S} G_{v}. Also, note that a particular cuspidal ∏ on G(𝔸) will appear on the right side of (2.4) only if it is distinguished, i.e., D(ϕ′) ≠ 0 for some ϕ′. Thus it is not surprising that a “linear independence of characters argument” yields the desired bijection of Theorem 1.1. For more details, we refer the reader to [Ye], and §4 below.

Since the most subtle part of the proof outlined above is Step II, we focus on this step in some detail in the sections below.

3. Step II: Subtracting off the Continuous Spectrum. Once we know that the relative trace formulas on H and G agree for matching f and f′, it follows (keeping in mind the previous discussion) that

(3.1) ∫∫ K_{f,0}(n_{1}, n_{2})ψ_{N}(n^{−1}n_{2}) dn_{1} dn_{2} − ∫∫ K_{f′,0}(n′, h)ψ_{N′}(n′) dn′ dh

= ∫∫ K_{f,cont}(n_{1}, n_{2})ψ_{N}(n^{−1}n_{2}) dn_{1} dn_{2} − ∫∫ K_{f′,cont}(n′, h)ψ_{N′}(n′) dn′ dh.

“Subtracting off the continuous spectrum” from the initial relative trace formula identity thus amounts to being able to prove that both sides of this last identity are identically zero.

According to (2.3) and (2.4), the left hand side of (3.1) reduces to the discrete expression

∑_{π} A(π, f_{S})(f′^{S})^{∨}(BC(t(π^{S})) − ∑_{Π} B(Π, f′_{S})(f′^{S})^{∨}(t(Π^{S})).

So the natural strategy should be to show that the right hand side of (3.1) can be written as the difference of two absolutely convergent expressions of the form

(3.2)  ∫ Φ(σ)(f′^{S})^{∨}(σ) dσ

with Φ(σ) an integrable function on the unramified unitary dual of G^{S}. In fact it turns out that the right-hand side of (3.1) contributes a discrete as well as continuous distribution (i.e., the continuous spectrums cancel each other only modulo an “unstable” piece of the cuspidal spectrum). Nevertheless, it is possible (after rearranging terms) to apply the trick of Langlands from [La2, p. 211]: since (3.1) amounts to an equality between an “atomic” and a “continuous” measure on the unramified unitary dual of G^{S}, both sides must be identically zero.

Consider for example the expression

(3.3)  ∫∫_{[N(F)\N(𝔸)]^{2}} K_{f,cont}(n_{1}, n_{2})ψ_{N}(n_{1}^{−1}n_{2}) dn_{1} dn_{2},

where

K_{f,cont}(x, y) = ∑_{χ={(M,μ)}} K_{χ}(x, y)

= ∑_{μ} ∑_{\{ϕ_{μ}\}} ∫_{−∞}^{∞} E(x, ρ(μ, it)(f)ϕ, it, μ)\overline{E(y, ϕ, it, μ)} dt.

Because the integration is over a compact domain, it is easy to compute that (3.3) equals

(3.4)  ∑_{μ ; unramified ; outside S} ∫_{−∞}^{∞}(f^{S})^{∨}(μ, it)c(f_{S}, μ, it) dt.

where (μ, it) denotes the conjugacy class in ^{L}G^{S} attached to Ind μ^{S}∥^{it},

c(f_{S}, μ, it) = ∑_{\{ϕ_{μ}\}} W (ρ(μ_{S}, it)(f_{S})ϕ, ψ, μ, it)\overline{W (ϕ, ψ, μ, it)},

and

W (ϕ, ψ, μ, it) = ∫_{N(F)\N(𝔸)} E(n, ϕ, it, μ)ψ_{N}(n) dn.

Moreover, because f^{S} “matches” f′^{S}, we have

(3.5)  (f^{S})^{∨}(μ, it) = f′^{S})^{∨}(BC(μ, it),

with (BC)(μ, it) = (μ′, it) and μ′(z) = μ(zz̄). Thus the contribution from K_{f,cont} to the right hand side of (3.1) is indeed of the form (3.2).

Now consider the relative trace formula expression

∫_{Z(𝔸)H(F)\H(𝔸)} ∫_{N′(F)\N′(𝔸)} K_{f′_{cont}}(n′, h)ψ_{N′}(n′) dn′ dh

corresponding to the quadratic extension. As we already hinted (and shall explain in the next section), this turns out to produce discrete as well as continuous sums of traces (f′^{S})^{∨}(σ).

4. Step II Continued: Periods of Truncated Eisenstein Series, and the Continuous relative trace over G. Our purpose here is to describe some of the difficulties involved in proving the following result:

Theorem 4.1. The contribution of the continuous spectrum to the relative trace formula for G is

∑_{\begin{matrix}μ′ \\ μ′=μ∘N_{E/F} \end{matrix}} ∫_{−∞}^{∞}(f′^{S})^{∨}(μ′, it)c_{1}(f′_{S}, μ′, it) dt + ∑_{\begin{matrix}μ′ \\ μ′≠μ∘N_{E/F} \end{matrix}} (f′^{S})^{∨}(μ′, 0)c_{2}(f′_{S}, μ′, 0).

Here both expressions are absolutely convergent,

∑_{μ′} |c_{2}(f′_{S}, μ′, 0)| < ∞, and ∑_{μ′} ∫_{−∞}^{∞} |c_{1}(f′_{S}, μ′, it)| dt < ∞.

As we already explained in the last section, this theorem is exactly what is needed to complete the proof of Theorem 1.1. Indeed, if we subtract the integral expression here from

∫∫ K_{f,cont}(n_{1}, n_{2})ψ_{N}(n_{1}^{−1}n_{2}) dn_{1} dn_{2}

in (3.1) we obtain (using(3.4) and (3.5)) the expression

(4.2)  ∑_{μ′} ∫_{−∞}^{∞}(f′^{S})(μ′, it)c_{∗}(f′_{S}, μ′, it) dt.

On the other hand, bringing the discrete sum in (4.1) over to the left-hand side of (3.1) yields the expression

(4.3) ∑_{π ; cuspidal} A(π, f_{S})(f′^{S})(BC(t(π^{S}))

∑_{Π cuspidal} B(Π, f′_{S})(f′^{S})^{∨}(t(Π^{S}))

− ∑_{μ≠μ∘N_{E/F}} (f′^{S})^{∨}(μ′, 0)c_{2}(f′_{S}, μ′, 0).

Hence we indeed can conclude (by the arbitrariness of (f′^{S})) that both (4.2) and (4.3) are identically zero.

Remark. For any given grossencharacter μ′ of E which does not factor through the norm map, let π(μ′) denote the cuspidal representation of H(𝔸) whose Langlands parameter t_{π_{v}} almost everywhere base change lifts to the Langlands parameter of Ind_{B′}^{G} μ′_{v}. These π(μ′) are the cusp forms constructed by Hecke and Maass (using either the theory of theta-functions or L-functions). The fact that (4.3) equals zero gives (by “linear independence of characters”) an alternate proof of the existence of such forms, as well as establishing the bijection asserted in Theorem 1.1.

Now let us return to Theorem 4.1, giving a concrete expression for the relative trace of R_{cont}(f′). As we have already noted—for α(T) sufficiently large,

∫_{Z(𝔸)H(F)\H(𝔸)} ∫_{N′(F)\N′(𝔸)} K_{f′_{cont}}(n′, h)ψ_{N′}(n′) dn′ dh

= ∫∫ Λ_{2}^{T} K_{f′,cont}(n′, h)ψ′_{N}(n′) dn′ dh.

Moreover, it follows from Propositions 2.3 and 2.5 of [Ja1] that this last expression equals

∑_{χ={(M′,μ′)}} ∫∫ Λ_{2}^{T} K_{χ}(n′, h) ψ_{N′}(n′) dn′ dh

and, ultimately, that

(4.4) ∫_{Z(𝔸)H(F)\H(𝔸)} ∫_{N′(F)\N′(𝔸)} K_{f′_{cont}}(n′, h)ψ_{N′}(n′) dn′ dh

= ∑_{μ′} ∫_{−∞}^{∞} ∑_{{ϕ′_{μ′} }}(∫_{Z(𝔸)H(F)\H(𝔸)} Λ^{T} E(h, ρ(μ′, it)(f′)ϕ′, it) dh)

× \overline{W (ϕ′, ψ, μ′, it)(e)} dt.

Thus our task is to obtain an explicit expression for the right-hand side of this last equation, as T → ∞ (i.e., for T sufficiently large …); in this we follow [Ja1].

Remark. Introducing a truncation operation facilitates the explicit calculation of the relative continuous spectrum of G but, as with the ordinary trace formula, also produces “discrete traces” from within the continuous spectrum, as we shall now explain.

In (4.4), the sum is over the (classes of) unitary characters μ′ of E^{∗}\𝔸_{E}^{x} inducing the Eisenstein series

E(g, ϕ′, μ′, s) = ∑_{γ∈B′(F)\G(F)} ϕ_{μ}(γg),

with ϕ_{μ} ∈ Ind μ′∥^{s}, and B′ = Res_{F}^{E} B = M′N′. In computing the periods

P(ϕ′, μ′, it) = ∫_{Z(𝔸)H(F)\H(𝔸)} Λ^{T} E(h, ρ(μ, it)(f)ϕ′, μ′, it) dh,

one appeals to the Bruhat-type decomposition

(*)  G(F) = B′(F)H(F) ⋃ B′(F)ηH(F),

where η in G(F) satisfies ηη̄^{−1} = w. (One can take η = ( \begin{matrix}\sqrt{τ} & 1 \\ −\sqrt{τ} & 1 \end{matrix} ) if \sqrt{τ} generates E.) The proposition below (see [JLai] and [Ja1]) is to be viewed as a “relative analogue” of the Langlands formula for the inner product of two truncated Eisenstein series.

Proposition 4.5. For each unitary character μ′ of E^{∗}\𝔸_{E}^{∗}, set

δ_{1}(μ′) = \{ \begin{matrix}m(F^{x}\𝔸_{F}^{1} ) & if μ′ |_{F^{x}\𝔸_{F}^{1}} ≡ 1; \\ 0 & otherwise, \end{matrix}

and

δ_{2}(μ′) = \{ \begin{matrix}m(𝔸_{F}^{∗} E^{x}\𝔸_{E}^{∗}) & if μ(a) ≡ μ(ā); \\ 0 & otherwise, \end{matrix}

Then for sufficiently large α(T) ,

P(ϕ′, μ′, it)

= δ_{1}(μ′)\{\frac{e^{it(T_{1}−T_{2})}}{it} ∫_{K} ϕ′(k) dk − \frac{e^{−it(T_{1}−T_{2})}}{it} ∫_{K} M(w, it)ϕ′(k) dk\}

+ δ_{2}(μ′) ∫_{T η(𝔸)\H(𝔸_{F})} ϕ′(ηh) dh

with

T_{η} = η^{−1}B′(F)η ⋂ H(F).

Remark. The δ_{1}-term here corresponds to the contribution from the “small-cell” part of (*), whereas the δ_{2}-term corresponds to the “big cell” (w ≠ e). Note that the second term is independent of the truncation parameter T. Also, all terms must be understood in the sense of meromorphic continuation from s in some right half-plane to s = it.

Sketch of the Proof. From the definition of Λ^{T}, and the formula for the constant term of E, one computes that

(4.6) Λ^{T} E(g, ϕ′, μ′, s) = ∑_{γ∈B′(F)\G(F)} ϕ′(γg)(1 − τ̂_{B}(H(γg) − T)

− M(w, s)ϕ′(γg)τ̂_{B}(H)(γg) − T).

(This is a special case of Arthur’s “second formula” for Λ^{T} E; see Lemma 4.1 of [A3].) The next step is to check that (for s in the domain of convergence of the Eisenstein series), the series

∑_{γ∈B′(F)\G(F)} |ϕ′(γg)| |1 − τ̂_{B}(H(γg) − T)| + |M(w, s)ϕ′(γg)|τ̂_{B}(H(γg) − T)

is integrable over Z(𝔸)H(F)\H(𝔸). This justifies the term by term integration of (4.6) over Z(𝔸)H(F)\H(𝔸); see [Ja1] §3 for the case of GL(3). Then the decomposition (*), together with Iwasawa’s decomposition for H(𝔸), ultimately yields the Proposition. Implicit here is the fact that for α(T) sufficiently large,

τ̂_{B}(H(ηh) − T) ≡ 0 for all h;

this requires some proof (see Proposition 7.2 of [Ja1] for a more general setting), and explains why the contribution from η is indeed independent of T.

In applying Proposition 4.5 to the calculation of

(4.1)  ∫∫ K_{f′,cont}(n′, h)ψ_{N′}(n′) dn′ dh,

let us first consider the contributions from characters μ′ with δ_{2}(μ′) ≠ 0. These are precisely the grossencharacters of E which factor thru the norm map from F. If in addition δ_{1}(μ′) = 0, then Proposition 4.5 implies that the contribution of such μ′ to (4.1) is

(4.7)  ∑_{μ′} ∫_{−∞}^{∞} ∑_{{ϕ′_{μ′} }} θ_{f}(μ′, it, ϕ′)W (ϕ′, ψ, μ′, it) dt

where

θ_{f}(μ′, it, ϕ′) = δ_{2}(μ′) ∫_{T_{η}(𝔸_{F})\H(𝔸_{F})} ρ(μ′, it)(f)ϕ′(ηh) dh.

Note we can rewrite (4.7) as

∑_{μ′} ∫_{−∞}^{∞}(f′^{S})^{∨}(μ′, it)C_{f′_{S}}(μ′, it) dt

with

C_{f′_{S}}(μ′, it) dt = ∑_{ϕ′} W (ϕ′, ψ, μ′, it)θ_{f′_{S}}(μ′, it, ϕ′).

Now suppose δ_{2}(μ′) = 0, i.e., μ′ does not factor thru the norm map from F. Then Proposition 4.5 implies that the contribution of each such μ′ to the relative trace (4.1) is

(4.8) ∫ ∑_{ϕ′} W (ϕ′, ψ, μ′, it)·

δ_{1}(μ′) \{\frac{e^{it(T_{1}−T_{2})}}{it} ∫_{K} fϕ′(h) dh − \frac{e^{−it(T_{1}−T_{2})}}{it} ∫_{K} M(w, it)fϕ′(k) dk\} dt,

with

fϕ′ = ρ(μ′, it)(f)ϕ′.

Notice the similarity here with the formula encountered for the ordinary trace formula term J_{χ}^{T} (f) in §4 of Lecture VII. There we computed the limit as T → ∞ by applying Arthur’s splitting formula for products of (G, M)-families. Here we can do the same thing, taking (as before)

c_{B}(it) = e^{2it(T_{1}−T_{2})} and cB̄(it) = e^{−2it(T_{1}−T_{2})}.

But now we set

d_{B}(it) = W (ϕ′, ψ, μ′, it) ∫_{K} fϕ′(k) dk

and

dB̄(it) = W (ϕ′, ψ, μ′, it) ∫_{K} M(w, it)(fϕ′)(k) dk.

It must of course be checked that this d(Λ) is also a (G, M)-family (not obvious!). The main task, though, is to compute

lim_{ε→0} lim_{T →∞} ∑_{ϕ′}(cd)_{M}(it)B(εt) dt,

with B(t) a compactly supported Schwartz function as in Lecture VII. Computing as before, we find that

d′_{G}(it) = d_{B}(0)(= dB̄(0)) = W (ϕ′, ψ, μ′, 0) ∫_{K} ρ(μ′, 0)(f)ϕ′(k) dk

with ϕ′ in Ind μ′, and—ultimately—that the above limit equals

∑_{ϕ} W (ϕ′, ψ, 0, μ′) ∫ ρ(0, μ′)(f′)ϕ′(k) dk.

Thus we find that the total contribution (to the relative trace formula) from characters μ′ of this form is

∑_{\begin{matrix}μ′ \\ μ′≠N∘μ \end{matrix}}(f′^{S})^{∨}(0, μ′)C(f′_{S}, μ′)

with

C(ψ, f′_{S}, μ′) = ∑_{ϕ} W (ϕ′, ψ, 0, μ′) ∫_{K} ρ(0, μ′_{S})(f′_{S})ϕ′(k) dk.

It remains to observe that when δ_{2}(μ′) ≠ 0 and δ_{1}(μ′) ≠ 0, the contribution to the relative trace formula is again just (4.5). There are apparent contributions

\frac{e^{2it(T_{1}−T_{2})}}{it} ∫ fϕ′(k) dk − \frac{e^{−2it(T_{1}−T_{2})}}{it} ∫ M(w, it)ϕ′(k) dk

from P(ϕ′, μ′, t), but they integrate to zero over ℝ by way of a Riemann–Lebesgue Lemma (just like the terms in cd_{M}(t) not belonging to the leading term Q = G in F(M) at the end of the last Lecture). We spare the reader the details.

Concluding Remarks. A host of natural problems present themselves in connection with any general development of the relative trace formula, especially as far as applications are concerned. In addition to the obvious problem of developing general local “matching” theorems (and related “fundamental lemmas”), there are these problems from the “spectral side”:

(1) Develop a general formula for the period of a truncated Eisenstein series, analogous to Langlands’ formula for the inner product of truncated Eisenstein series;

(2) Apply a general theory of (G, M)-families and smoothing functions B(λ) to the calculation of the contribution of the “continuous spectrum” to the relative trace formula, and

(3) Prove in general that the resulting spectral expressions are sufficiently convergent to justify subtracting them off from the full relative trace formula. For the case of GL(2), this can done by direct estimates on the intertwining operators, but already for GL(3) such a direct approach is problematic.

References

[Ja1] Jacquet, H., The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension, Israel Journal of Math. 89, 1–59.

[Ja2] , Relative Kloosterman integrals for GL(3), II, Canad. J. Math. 44 (1992), 1220– 1240.

[JLai] Jacquet, H. , and Lai, J., A relative trace formula, Comp. Math. 54 (1985), 243–310.

[JLR] Jacquet, H. , Lai, K., and Rallis, S., A trace formula for symmetric spaces, Duke Math. J. 70 (1993), no. 2, 305–372.

[JaYe] Jacquet, H. , and Ye, Y., Une remarque sur le changement de base quadratic, C.R.A.S. Paris 311 (1990), 671–676.

[Mao] Mao, Z., Relative Kloosterman integrals for GL(3), III, Canad. J. Math. 45 (1993), no. 6, 1211–12430.

[Ye] Ye, Y., Kloosterman integrals and base change for GL(2), J. Reine Angew. Math. 400 (1989), 57–121.

LECTURE IX. APPLICATIONS OF PALEY–WIENER, AND CONCLUDING REMARKS87

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

In Lecture VII, we explained how (G, M)-families arose in Arthur’s explicit calculation of the general spectral terms J_{χ}(f). We also alluded to why a Paley–Wiener theorem was needed to complete this calculation, and promised to return to it. In fact, this Paley–Wiener theorem is also crucial in handling other convergence problems related to the trace formula. So our purpose now is to finally touch on these matters, albeit briefly. This being the last lecture, we shall also conclude with a few remarks orienting the reader to some papers of Arthur’s that we so far haven’t even mentioned.

1. The Paley–Wiener Theorem. The Fourier transform of a compactly supported smooth function f on ℝ extends to an entire function F (Λ) on ℂ, and satisfies a well-known growth condition, namely that there exists a constant N (depending on the support of f) such that

sup{|F (Λ)|e^{−N| Re(Λ)|}(1 + | Im(Λ)|)^{n}} < ∞

for any integer n. According to the classical Paley–Wiener Theorem, these properties characterize the image of the Fourier transform on the space of such test functions f.

For a real reductive group G, let C_{c}^{∞}(G, K) denote the “Hecke algebra” of Kfinite compactly supported smooth functions. Then for any f in C_{c}^{∞}(G, K) and irreducible admissible representation π of G (on a Banach space U_{π}), the Fourier transform of f at π is defined to be the operator

π(f) = ∫_{G} f(x)π(x) dx.

The resulting function

π → π(f)

has as its domain the set of irreducible (admissible) representations of G, and for any (π, U_{π}) takes values in the space of operators on U_{π}. The purpose of the (operator) Paley–Wiener Theorem of [A10] is to characterize which functions

π ↦ F (π)

are of the form π ↦ π(f) for some f.

More precisely, let us (following Arthur) fix a minimal parabolic subgroup B of G with Langlands decomposition N_{0}A_{0}M_{0}^{1}, and consider the nonunitary principal series of induced representations ρ_{B}(σ, Λ), indexed by quasi-characters Λ of A_{0} and irreducible representations σ of M_{0}^{1}. By a well-known theorem of Harish-Chandra, any π is equivalent to a subquotient of some ρ_{B}(σ, Λ). This means that π(f) will be completely determined by the Fourier transform

f̂: (σ, Λ) → f̂_{B}(σ, Λ) = ρ_{B}(σ, Λ)(f).

What should the image of this Fourier transform be?

Arthur defines P W (G, K) to be the space of functions

F: (σ, Λ) ↦ F_{B}(σ, Λ),

which are entire, K-finite (in a sense I shall not explain here), and satisfy the growth condition

(1.1)  sup_{σ,Λ}{∥F (σ, Λ)∥e^{−N∥ Re(Λ)∥}(1 + ∥ Im(Λ)∥)^{n}} < ∞,

for some N (and all n). Moreover, there is another, more complicated condition for F (σ, Λ) which comes from the various intertwining maps between principal series. Namely, if there is a relation

∑_{k=1}^{m} D_{k}(ρ_{B}(σ_{k}, Λ_{k})(g)u_{k}, v_{k}) = 0

valid for all g in G, where D_{k} is a differential operator (acting through the variable Λ, and u_{k}, v_{k} are vectors in the space of ρ(σ_{k}, Λ_{k}), then F (σ, Λ) must also satisfy the relation

(1.2)  ∑_{k=1}^{m} D_{k}(F (σ_{k}, Λ_{k})u_{k}, v_{k}) = 0.

(Such relations exist commonly, but are difficult to characterize explicitly; indeed, a listing of all such relations would be tantamount to a complete knowledge of the irreducible subquotients of the principal series.)

The main result of [A10] is that

f → f̂(σ, Λ)

provides an algebra isomorphism from C_{c}^{∞}(G, K) onto PW(G, K). The consequence of this result which is actually needed for the trace formula concerns the notion of Paley–Wiener multipliers.

By a “multiplier” of C_{c}^{∞}(G, K) is meant a linear operator C on C_{c}^{∞}(G, K) such that

C(f ∗ g) = C(f) ∗ g = f ∗ C(g)

for all f, g in C_{c}^{∞}(G, K) . (The algebra of all such “multipliers” coincides with End_{𝔘(G)}(C_{c}^{∞}(G, K)). The Paley–Wiener multipliers are constructed as follows. Set

(1.3)  𝔥 = i𝔥_{K} ⊕ 𝔥_{0},

where 𝔥_{K} is a fixed Cartan subalgebra of the Lie algebra of K ∩ M_{0}, and 𝔥_{0} is the Lie algebra of the split component of M_{0}. (Here M_{0} is the Levi component of a fixed minimal parabolic subgroup of G.) If 𝔊 is the Lie algebra of G, then 𝔥_{ℂ} is a Cartan subalgebra of 𝔊_{ℂ}, and 𝔥 ⊂ 𝔥_{ℂ} is invariant under the Weyl group W of (𝔊_{ℂ}, 𝔥_{ℂ}). Let E(𝔥)^{W} be the algebra of compactly supported distributions on 𝔥 which are invariant under W. Then for any γ in E(𝔥)^{W}, the Fourier–Laplace transform

γ̂(ν), ν ∈ 𝔥_{ℂ}^{∗}

is an entire function on 𝔥_{ℂ}^{∗} which is W -invariant and satisfies a growth estimate of the form

(1.4)  sup_{ν∈𝔥_{ℂ}^{∗}}(|γ̂(ν)|e^{−N_{γ}∥ Re(ν)∥}(1 + ∥ Im ν∥^{−n_{γ}} ) < M

for some integers N_{γ} and n_{γ}. According to the result below, these γ̂ provide us with multipliers C_{γ} (called Paley–Wiener multipliers).

Theorem 1.5 (see Thm. 4.2 of [A10]). For each distribution γ in E(𝔥)^{W}, and f in C_{c}^{∞}(G, K), there is a unique f_{γ} in C_{c}^{∞}(G, K) such that

(1.5)  π(f_{γ}) = γ̂(ν_{π})π(f)

for any irreducible admissible representation π. (Here {ν_{π}} is the W -orbit in h_{ℂ}^{∗} associated to the infinitesimal character of π.) In particular, if we define

C_{γ}(f) = f_{γ},

then the map

γ → C_{γ}, γ ∈ E(𝔥)^{W} ,

is a homomorphism from the algebra E(𝔥)^{W} to the algebra of multipliers

End_{𝔘(G)}(C_{c}^{∞}(G, K)).

Remarks. (i) This Theorem is almost an immediate consequence of the Paley– Wiener Theorem described earlier. Indeed, the uniqueness is clear (since (1.5) completely determines the Fourier transform of f_{γ}). On the other hand, the existence of f_{γ} results from checking that the function

F_{B}(σ, Λ) = γ̂(ν_{σ} + Λ)f̂_{B}(σ, Λ)

belongs to PW(G, K), and hence is the Fourier transform of some f_{γ}; then since π(f_{γ}) is given by the action of ρ(σ, Λ)(f_{γ}) on the appropriate subquotient, it follows that

π(f_{γ}) = γ̂(ν_{σ} + Λ)π(f) = γ̂(ν_{π})π(f),

as required.

(ii) That C_{γ}(f) = f_{γ} defines a “multiplier” of C_{c}^{∞}(G, K) follows formally from (1.5): C_{γ}(f ∗ g) = C_{γ}(f) ∗ g, since

π((f ∗ g)_{γ}) = γ̂(ν_{π})π(f ∗ g)
= γ̂(ν_{π})π(f)π(g)
= π(f_{γ})π(g).

(iii) This Paley–Wiener theorem may be viewed as an analogue for real groups of the p-adic results of [BDK] and [Ro3].

2. Applications of Paley–Wiener to the Calculation of J_{χ}(f).. Recall how J_{χ}^{T} (f) was defined (for sufficiently large T) to be the integral over the diagonal of the truncated kernel function Λ_{2}^{T} (K_{χ}(x, y). Using the explicit formula for Λ_{2}^{T} (K_{χ}(x, y) in terms of Eisenstein series, we then wrote

(2.1)  J_{χ}^{T} (f) = ∑_{P} ∑_{σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f) dλ,

where

Ψ_{σ}^{T} (λ, f) = \frac{1}{n(A_{P})} tr(Ω_{χ,σ}^{T}(P, λ)ρ(σ, λ)(f)),

and Ω_{χ,σ}^{T}(P, λ) is the operator on the space of ρ(σ, λ) defined by

(Ω_{χ,σ}^{T}(P, λ)ϕ′, ϕ) = ∫_{Z(𝔸)G(F)\G(𝔸)} Λ^{T} E(x, ϕ′, λ)\overline{Λ^{T} E(x, ϕ, λ)} dx.

We stress that the integral formula (2.1) for J_{χ}^{T} (f) is absolutely convergent, and that J_{χ}^{T} (f) is known to be polynomial in T. The problem is to explicitly compute J_{χ}^{T} (f) for a conveniently chosen value T_{0} of T.

At the end of Lecture IV, we used Arthur’s general theory to derive an explicit formula for J_{χ}^{T} (f) in the case of G = GL(2). The strategy consisted of three steps:

(1) plug into (2.1) Langlands’ explicit formula for Ω_{χ,σ}^{T}(P, λ) in terms of intertwining operators;

(2) use the theory of (G, M)-families to rewrite Ψ_{σ}^{T} (λ, f) in a form still more amenable to integration over dλ; and

(3) compute the resulting integrals over λ (and show that some of the contributions are negligible as T → ∞).

It was in this last step that we resorted to a Riemann–Lebesgue Lemma whose application is best justified by way of Paley–Wiener multipliers; indeed these multipliers allow for the insertion of a compactly supported function B(λ) into the formula for J_{χ}T, as explained in Theorem 2.3 below.

For general G, there is an additional, more fundamental reason for requiring the Paley–Wiener Theorem in the calculation of J_{χ}(f). It derives from the fact that Langlands’ explicit formula for Ω_{χ,σ}^{T}(P, λ) is not valid for Eisenstein series induced from non-cuspidal σ. (Equivalently, it is not valid when P lies outside P_{χ}, the set of parabolics attached to the collection of pairs (M_{B}, r_{B}) describing the cuspidal datum χ.) In these cases, Arthur proved “only” that the “nice” formula for Ω_{χ,σ}^{T}(P, λ) is an asymptotic one for large T, valid uniformly only on compact subsets of λ.

More precisely, let ω_{χ,σ}^{T}(P, λ) denote the operator on ρ(σ, λ) defined explicitly in terms of intertwining operators as the value at λ′ = λ of

∑_{P_{1}} ∑_{t,t′} M_{P_{1}|P} (t, λ)^{−1}M_{P_{1}|P} (t′, λ′)\frac{e^{(t′λ′−tλ)(T)}}{θ_{P_{1}}(t′λ′ − tλ)}

(cf. formula (2.1) of Lecture VII for complete details). Then (cf. Corollary 9.2 of [A9]) the difference between (Ω_{χ,σ}^{T}(P, λ)ρ(λ, σ)(f)ϕ, ϕ) and (the “nicer” expression) (ω_{χ,σ}^{T}(P, λ)ρ(λ, σ)(f)ϕ, ϕ) is bounded in absolute value by

(2.2)  r(λ)∥ϕ∥^{2}e^{−ϵ∥T ∥},

where ϵ > 0, and r(λ) is a locally bounded function on i𝔞_{P}^{∗}. Thus the integrals of these expressions over all of i𝔞_{P}^{∗} /i𝔞_{G}^{∗} need not approach one another as T → ∞. In short, the substitution of ω_{χ,σ}^{T} for Ω_{χ,σ}^{T} in (2.1) is simply not justified.

Example. For G = GL(2), we explicitly derived Arthur’s estimate(2.2) in case χ = {(M, μ)}, with μ = μ^{−1}, and P = G. Namely, in the proof of Prop. 2.7 in Lecture IV, we computed the difference of these expressions to be Me^{−2(T_{1}−T_{2})}. Of course, in this case the problem of uniformity alluded to above is a priori absent (the variable λ being constrained to lie in the space i𝔞_{P}^{∗} /i𝔞_{G}^{∗} = {0} …).

In general, the way out of this problem, like the problem of showing certain terms to be “negligible” in step (3) above, is by way of Arthur’s Paley–Wiener Theorem, as we shall now (finally) explain.

For any B in S(i𝔥^{∗}/i𝔞_{G}^{∗})^{W}, and irreducible unitary representation σ on M(𝔸), define

B_{σ}(λ) = B(iY_{σ} + λ),

Here Y_{σ} is the imaginary part of the orbit {ν_{σ}} in 𝔥_{ℂ}^{∗} associated to the infinitesimal character of σ_{∞} (and we have fixed an embedding of any 𝔞_{P}^{∗} in 𝔥^{∗}). Recall that {ν_{σ} + λ} is also the orbit in 𝔥_{ℂ}^{∗} associated to the infinitesimal character of the induced representation ρ(σ_{∞}, λ) of G_{∞}.

Theorem 2.3 (see Thm. 6.3 of [A5]). Suppose B in S(i𝔥^{∗}/i𝔞_{G}^{∗})^{W} is such that B(0) = 1. For any ϵ > 0, write B^{ϵ} for the function B^{ϵ}(ν) = B(ϵν). Then

J_{χ}^{T} (f) = lim_{ϵ→0} lim_{T →∞} ∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f)B_{σ}^{ϵ}(λ) dλ.

Remarks. (i) The importance of this result is that it allows us to invoke the asymptotic formula for Ω_{χ,σ}^{T}. Indeed, if we take B in the Theorem to be compactly supported, then we can indeed substitute ω_{χ,σ}^{T} for Ω_{χ,σ}^{T} and write

(2.4)  J_{χ}^{T} (f) = lim_{ϵ→0} lim_{T →∞} ∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} tr(ω_{χ,σ}^{T}(P, λ)ρ(λ, σ)(f))B_{σ}^{ϵ}(λ) dλ

(since the error term (2.2), multiplied by such a compactly supported B, will indeed approach 0 uniformly in λ).

(ii) For large T, the expression

∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f)B_{σ}(λ) dλ

is asymptotic to a polynomial P^{T} (B) in T. So Theorem 2.3 really says

J_{χ}^{T} (f) = lim_{ϵ→0} P^{T} (B^{ϵ}),

or

J_{χ}^{T} (f) = lim_{ϵ→0} lim_{T →∞} ∑_{P,σ} ∫ Ψ_{σ}^{T} (λ, f)B_{σ}^{ϵ}(λ) dλ,

where the limit in T (as in the statement of the Theorem) is interpreted as the polynomial P^{T} (B^{ϵ}) which is asymptotic to the given function as T approaches ∞. On the other hand, since

lim_{ϵ→0} B_{σ}^{ϵ}(λ) = lim_{ϵ→0} B(ϵ(iY_{σ} + λ)) = B(0) = 1,

the dominated convergence theorem implies

J_{χ}^{T} (f) = ∑_{P,σ} ∫ Ψ_{σ}^{T} (λ) dλ  (for T large)

= lim_{ϵ→0} ∑_{P,σ} ∫ Ψ_{σ}^{T} (λ)B_{σ}^{ϵ}(λ) dλ (T large)

= lim_{T →∞} lim_{ϵ→0} ∑_{P,σ} ∫ Ψ_{σ}^{T} (λ)B_{σ}^{ϵ}(λ) dλ

(with lim_{T →∞} interpreted as above). Thus all of the work of [A5] essentially boils down to a justification of the interchange of order of two limits (over ϵ and T)!

(iii) (Concerning the Proof of Theorem 2.3). The basic idea of the proof is explained clearly on pp. 1258–60 of [A5]. The starting point is an application of the Paley–Wiener multiplier Theorem 1.5 to the archimedean component of any K-finite function f in C_{c}^{∞}(G(𝔸)). For any γ in E(𝔥)^{W}, this yields a function f_{γ} in C_{c}^{∞}(G(𝔸)) such that

Ψ_{σ}^{T} (λ, f_{λ}) = γ̂(ν_{σ} + λ)Ψ_{σ}^{T} (λ, f).

Using this relation, we write

(2.5)

J_{χ}^{T} (f_{λ}) = ∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} γ̂(ν_{σ} + λ)Ψ_{σ}^{T} (λ, f) dλ  (for T large)

= ∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f)(∫_{𝔥} γ(H)e^{(ν_{σ}+λ)(H)} dH) dλ

= ∫_{𝔥}(∑_{P,σ} Ψ_{σ}^{T} (H)e^{ν_{σ}(H)})γ(H) dH

where

Ψ_{σ}^{T} (H) = ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f)e^{λ(H)} dλ.

In particular, if we fix H in 𝔥, and set

γ^{H} = |W |^{−1} ∑_{s∈W} γ_{s^{−1}H}

with γ_{s^{−1}H} the Dirac measure at s^{−1}H, then (2.5) implies

(2.6)  J_{χ}^{T} (f_{γ^{H}}) = |W |^{−1} ∑_{s∈W} ∑_{P,σ} Ψ_{σ}^{T} (s^{−1}H)e^{ν_{σ}(s^{−1}H)}

Note here that, for each value of H, the right hand side of (2.6) is a polynomial in T (since the left hand side is). Thus we have constructed a family of polynomials in T, call them p^{T} (H), whose value at H = 0 is exactly J_{χ}^{T} (f) (since then f_{γ^{H}} = f).

Now to compute p^{T} (H) at 0, the natural thing to do is to integrate this function against an arbitrary Schwartz function β(H). By the Plancherel Theorem for 𝔥, the resulting inner product ∫ p^{T} (H)β(H) dH can be replaced by one over i𝔥^{∗}/i𝔞_{G}^{∗} involving the Fourier transform

B(ν) = ∫ β(H)e^{ν(H)} dH.

In particular, J_{χ}^{T} (f) should be obtained by having β approximate the Dirac measure at 0, i.e., by using

β_{ϵ}(H) = ϵ^{− dim(𝔥)}β(ϵ^{−1}H)

with ϵ small, and ∫ β(H) dH = 1; Equivalently, this brings into play

B^{ϵ}(ν) = B(ϵν)

with B(0) = 1, and this is roughly how one ends up with the sought-after formula

J_{χ}^{T} (f) = lim_{ϵ→0} ∑_{P,σ} ∫_{i𝔞_{P}^{∗} /i𝔞_{G}^{∗}} Ψ_{σ}^{T} (λ, f)B_{σ}^{ϵ}(λ) dλ.

The only problem is that p^{T} (H) is not actually tempered, and hence cannot be integrated against any β! Thus a lengthy detour is required in Sections 4–6 of [A5], involving additional polynomials and analysis (but nothing approaching either the Paley–Wiener theorem, or the theorem on the polynomial nature of J_{χ}^{T} (f)).

3. A Final Formula for ∑_{χ} J_{χ}^{T} (f). At the end of Lecture VII, we explained how, for GL(2), the formula

J_{χ}^{T} (f) = lim_{ϵ→0} lim_{T →∞} ∫_{−∞}^{∞} tr(ω_{χ}^{T} (t)ρ(μ, it)(f)B(ϵt) dt

could be computed explicitly using (G, M) families. Now, thanks to formula 2.4, we have a similar strategy for computing J_{χ}^{T} (f) in general, and the end result is a formula directly generalizing the explicit formula

J_{χ}^{T} (f) = (T_{1} − T_{2}) ∫_{−∞}^{∞} tr(ρ(μ, it)(f)) dt + ∫_{−∞}^{∞} tr(M(−it)M′(it)ρ(μ, it)(f) dt

+ \frac{1}{4} tr(M(0)ρ(μ, 0)(f)) + μ(f)τ(G).

(Recall that the “limit in T ” is to be interpreted as the polynomial asymptotic to the given integral as T approaches ∞.) N.B. In Lecture IV, we were interested only in the constant term of the polynomial J_{χ}^{T} (f), i.e., J_{χ}^{T} (f) = J_{χ}^{0}(f). In Arthur’s general theory, one fixes on a convenient point T_{0} (which in many “classical” situations is just zero; see Lemma 1.1 of [A4]), and sets

J_{χ}(f) = J_{χ}^{T_{0}}(f).

Theorem 3.2 (see Theorem 8.2 of [A6]). Suppose f ∈ C_{c}^{∞}(Z(𝔸)\G(𝔸)). Then J_{χ}^{T} (f) equals the sum over M ∈ L(M_{0}), L ∈ L(M), σ on M(𝔸) (compatible with χ), and w in W^{L}(𝔞_{M})_{reg} of the product of

|W_{0}^{M}| |W_{0}|^{−1}| det(w − 1)_{𝔞_{M}^{L}})|^{−1}

with

∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} |P(M)|^{−1} ∑_{P ∈P(M)} tr(M_{L}(P, λ)M(P, w)ρ(σ, λ)(f)) dλ.

Remarks. (i) The exact definition of each of the objects appearing here is to be found in Sections 4 through 8 of [A6]. But the point is that they are all more or less familiar objects related to the intertwining operators M_{Q|P} discussed in Lecture VII. For GL_{2}, there will be only three terms in the sum above: one for M = L = M_{0}, with w = e and σ = μ a character of M_{0} (this gives the term in (3.1) with the logarithmic derivative); one for M = M_{0},L = G, w the non-trivial Weyl element, and σ a character of M_{0} (this gives the term involving tr(M(0)ρ(μ, 0)(f))); and one with M = L = G, w = e, and μ the character μ(det g) of G(𝔸) (this gives the contribution μ(f)τ(G)).

(ii) The proof of Theorem 3.2 consists of two non-trivial steps. First one uses (G, M) families to carry out the explicit computation of

val_{T =T_{0}} (lim_{ϵ→0} “ lim_{T →∞} ” ∑_{P} ∑_{σ} ∫ ω_{σ,χ}^{T}(λ, f)B_{σ}(ϵλ) dλ) ,

where ω_{σ,χ}^{T}(λ, f) is a sum of special values of functions e_{M}(Λ) for particular (G, M)- families e(Λ) = c(Λ)d(Λ) (involving the intertwining operators M_{Q|P}); this is carried out in Sections 3 through 5 of [A6], and the end result is a sum of terms of the form

lim_{ϵ→0} ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∑_{P ∈P(M)} tr(M_{L}(P, λ)M(P, w)ρ(λ, σ)B_{σ}^{ϵ}(λ)d

with B in C_{c}^{∞}(i𝔥^{∗}/i𝔞_{G}^{∗})^{W} equal to 1 at 0. Clearly the second step must be to show that this (limit and) test function can be made to disappear!

Equivalently, by the dominated convergence theorem, one must show that the sum

∑_{σ} ∫_{i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∥M_{L}(P, λ)ρ(λ, σ)(f)∥ dλ

is finite. This is carried out in Sections 6 through 8 of [A6], using normalized intertwining operators, and estimates on the scalar functions defining these normalizations. (Some assumptions are made on these normalizations at p-adic places, but at least for GL(n) these assumptions are now proved; see [Sha1] and [Sha2]).

Finally, having reached the exalted plateau which Theorem 3.2 represents, we should mention that it is really just the beginning! In particular it is not yet known if the explicit spectral expansion

(3.3)  ∑_{χ} J_{χ}(f) = ∑_{χ} ∑_{\begin{matrix}M,L \\ w \end{matrix}} ∑_{σ} ∫_{λ∈i𝔞_{L}^{∗}/i𝔞_{G}^{∗}} ∑_{P} tr(M_{L}^{∗}(P, λ, σ, w, f) dλ

converges absolutely as a multiple integral. (The problem is that the estimates on the normalizing factors alluded to above are not uniform in χ!).

In other words, we know that

∑_{χ} |J_{χ}(f)| < ∞,

and that

∑ ; σ∈∏(M(𝔸))_{χ} ∫ ∥M_{L}(P, λ)ρ(λ, σ)(f)∥ dλ < ∞,

but not that this last sum can be summed over all χ. This lack of knowledge causes problems in applying the trace formula, say to base change, where on needs absolutely convergent expressions (in order to use tricks like Langlands’ to subtract off the non-discrete spectrum). In order to get around this problem, Arthur proves certain estimates on the convergence of

∑_{χ} |J_{χ}(f)|

which may be viewed as a weak form of the conjectured absolute convergence of the multiple integrals (3.3). This involves the infinitesimal characters of the infinite components σ_{∞} of the representations σ appearing in the spectral expansion, and an application of the Paley–Wiener multiplier theorem. The end result is a technique called “separation of infinitesimal characters via multipliers” which allows one to obtain the required equality of discrete traces from the matching of the full traces; details appear in [A12], [AC], and [La3].

4. Related Works. Here are some of the topics from Arthur’s work which we have ignored till now.

In comparing trace formulas for different groups (for example, in proving base change for GL_{n}), one wants to rewrite the basic formula

(4.1)  ∑_{𝔬} J_{𝔬}(f) = ∑_{χ} J_{χ}(f)

in invariant form, i.e., as an identity

(4.2)  ∑_{𝔬} I_{0}(f) = ∑_{χ} I_{χ}(f),

where each of the terms is an invariant distribution. The reason for this is that one needs to assert the equality of two trace formulas for “associated” (or “matching”) functions, given only by their orbital integrals (which are invariant distributions …). Such an invariant trace formula is obtained already in [A4], where a subtle rearrangement and modification of the terms J_{𝔬}(f) and J_{χ}(f) yields the new collection of invariant distribution I_{𝔬}(f) and I_{χ}(f) appearing in (4.2). However, this is only the “coarse” form of the invariant trace formula, analogous to the “coarse” formula ∑ J_{𝔬}(f) = ∑ J_{χ}(f). To get the “fine” expansion of the invariant trace formula requires the papers [A11] and [A12]

Explicit formulas for I_{𝔬}(f) are obtained from the analogous fine expansions of J_{𝔬}(f) discussed in [A13] and [A14]. (In Lecture IV we described the exact form of J_{0}(f)—for unramified 𝔬—in terms of weighted orbital integrals; the extreme opposite case—of 𝔬 corresponding to {1}—is the subject matter of [A13], and the general case—a mixture of these extremes—is handled in [A14].) On the other hand, explicit formulas for I_{χ}(f) are obtained by another delicate mixture of the techniques of Paley–Wiener multipliers and (G, M)-families, somewhat analogously to the case of ∑ J_{χ}(f).

Remarks. (i) In a very recent work, Labesse has explained how one can apply the trace formula to the base change problem for GL(n) without first putting the trace formula in invariant form. This gives an alternate, simpler approach to [AC]; see [La3].

(ii) Recall from our discussion of Jacquet’s relative trace formula that no truncation operator was needed to define the “relative” distributions J_{𝔬}(f) and J_{χ}(f); hence these terms are already invariant!

(iii) Outside the scope of these “introductory” Lectures are the works of Kottwitz and others on the stable trace formula (see [Kot2] and [La5]), and the more recent papers of Arthur on unipotent representations, A-packets, etc. (see [A15], [A16]).

(New) References

[A15] Arthur, J., On some problems suggested by the trace formula, Lie Group Representations II,, Lecture Notes in Mathematics, vol. 1041, Springer-Verlag, 1984, pp. 1–49.

[A16] , Unipotent automorphic representations: global motivations, Automorphic Forms, Shimura Varieties and L-functions, vol. 1, Academic Press, 1990, pp. 1–75.

[Kot2] Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365–399; Duke Math. J. 51 (1984), 611–650.

[La5] Langlands, R. P., Les débuts d’une formule des traces stable, Publ. Math. de L’Univ. de Paris VII, vol. 13, 1983.

[Ro3] Rogawski, J., The trace Paley–Wiener Theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), no. 1, 215–229.

References for All Lectures

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

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

[A3] , A trace formula for reductive groups, II: Applications of a truncation operator, Compositio Math., vol. 40, Kluwer Acad. Publ., Dordrecht, 1980, pp. 87–121.

[A4] , A trace formula in invariant form, Ann. of Math. 113 (1981), 1–74.

[A5] , On a family of distributions obtained from Eisenstein series, I: Application of the Paley–Weiner theorem, Amer. J. Math. 104 (1982), no. 6, 1243–1288.

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

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

[A8] , The characters of discrete series as orbital integrals, Invent. Math. 32 (1976), 205–261.

[A9] , On the inner product of truncated Eisenstein series, Duke. Math. J. 49 (1982), no. 1, 35–70.

[A10] , A Paley–Wiener Theorem for real reductive groups, Acta Math. 150 (1983), 1–89.

[A11] , The invariant trace formula, I: Local theory, J. Amer. Math. Soc. 1 (1988), no. 2, 323–383.

[A12] , The invariant trace formula, II: Global theory, J. Amer. Math. Soc. 1 (1988), no. 3, 501–554.

[A13] , A measure on the unipotent orbit, Canad. J. Math. 37 (1985), no. 6, 1237–1274.

[A14] , On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986), no. 1, 179–214.

[A15] , On some problems suggested by the trace formula, Lie Group Representations II, Lecture Notes in Mathematics, vol. 1041, Springer-Verlag, 1984, pp. 1–49.

[A16] , Unipotent automorphic representations: global motivations, Automorphic Forms, Shimura Varieties and L-functions, vol. 1, Academic Press, 1990, pp. 1–75.

[AC] Arthur, J. and Clozel, L., Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud. No. 83, Princeton University Press, Princeton, NJ, 1975.

[BDK] Bernstein, J., Deligne, P., and Kazhdan, D, Trace Paley–Weiner theorem for reductive p-adic groups, J. Anal. Math. 47 (1986), 180–192.

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[Go2] , The Spectral Decomposition of Cusp Forms, Proc. Sympos. Pure Math., vol. IX, American Mathematical Society, Providence, RI, 1966, pp. 225–234.

[Ja1] Jacquet, H., The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension, Israel Journal of Math. 89, 1–59.

[Ja2] , Relative Kloosterman integrals for GL(3), II, Canad. J. Math. 44 (1992), 1220– 1240.

[JLai] Jacquet, H., and Lai, J., A relative trace formula, Comp. Math. 54 (1985), 243–310.

[JLR] Jacquet, H., Lai, K., and Rallis, S., A trace formula for symmetric spaces, Duke Math. J. 70 (1993), no. 2, 305–372.

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[Kot] Kottwitz, R., Tamagawa Numbers, Ann. of Math. 127 (1988), 629–646.

[Kot2] Kottwitz, R., Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365–399; Duke Math. J. 51 (1984), 611–650.

[Lai] Lai, K. F., Tamagawa Numbers of Reductive Algebraic Groups, Compositio Math., vol. 41, Kluwer Acad. Publ., Dordrecht, 1980, pp. 153–188.

[Lab1] Labesse, J.-P., La formules des traces D’Arthur-Selberg, Seminaire Bourbaki, exposé 636, November 1984.

[Lab2] , The present state of the trace formula, Automorphic Forms, Shimura Varieties, and L-functions (L. Clozel and J. Milne, eds.), Perspectives in Math., vol. 10, Academic Press, Princeton, NJ, 1990, pp. 211–226.

[Lab3] , Non-invariant base change identitites, Preprint (Fall 1994).

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[La2] , Base Change for GL(2), Ann. of Math. Stud., vol. 96, Princeton University Press, Princeton, NJ, 1980.

[La3] , The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, RI, 1966, pp. 143–148.

[La4] , Eisenstein series, the trace formula, and the modern theory of automorphic forms, Number Theory, Trace Formula and Discrete Groups (Selberg Volume), Academic Press, 1989.

[La5] , Les débuts d’une formule des traces stable, Publ. Math. de L’Univ. de Paris VII, vol. 13, 1983.

[Mao] Mao, Z., Relative Kloosterman integrals for GL(3), III, Canad. J. Math. 45 (1993), no. 6, 1211–1230.

[MW] Moeglin, C., and Waldspurger, J.-L., Décomposition Spectrale et Séries d’Eisenstein, Progress in Math. Series, vol. 113, Birkhäuser Verlag, 1994.

[Ro1] Rogawski, J., Representations of GL(n) and Division Algebras on a p-adic Field, Duke. Math. J. 50 (1983), no. 1, 161–196.

[Ro2] , Automorphic Representations of Unitary Groups in Three Variables, Ann. of Math. Stud., vol. 123, Princeton University Press, Princeton, NJ.

[Ro3] Rogawski, J., The trace Paley–Wiener Theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), no. 1, 215–229.

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Stephen Gelbart

Weizmann Institute of Science Theoretical Math Rehovot 76100, Israel mtgelbar@weizmann.weizmann.ac.il