Princeton University Princeton, N.J.

Dear Godement,

I do not know if this letter will serve as a report on Jacquet’s thesis. If a shorter statement or one of a different nature is necessary I would be willing to try to write it. In any case I wanted to write down as well as I could in the time available my thoughts on reading the thesis. In particular there are a number of problems suggested by the results of Jacquet and I would like to know if he has done anything with them.

I would like to see everything done in terms of general reductive groups for it would probably be useful to solve the technical problems which would arise. Consider first the question of convergence. It appears that the local reduction theory, as developed by Bruhat and Tits, is now in sufficiently good shape that the method of / 2 in the thesis can be used to reduce the question of convergence to the rank-one case. Let G be a reductive group over K with minimal parabolic subgroup P for which, when K = ℚ_{p}, conjecture I of Bruhat’s Boulder paper is satisfied. (It will be convenient to use his notation.) If Ū is the unipotent radical of the parabolic subgroup opposed to P one has to prove that

∫Ū_{K} L(gū, λ + ρ) dū

converges when Re(λ, α) > 0 for the simple roots of A. Here L(g, λ) is the obvious generalization of the corresponding function in Jacquet. Harish-Chandra has shown in his first paper on spherical functions that this is so when K = ℝ. There are two parts to his proof. He first studies the asymptotic behaviour of spherical functions; then he shows that the convergence of the integral is implied by the nature of the asymptotic behaviour of a particular spherical function. Although there would be some difficulties it appears that the second step can be carried through without major changes when K = ℚ_{p}. The first step is perhaps rather easier when K = ℚ_{p}.

Let H be the algebra of compactly supported functions on G bi-invariant under K. Let f ↦ f̃ be the homomorphism of H into the group algebra of A_{K}/A_{O} defined by

f̃(a) = L(a, −ρ) ∫_{U_{K}} f(au) du .

Presumably Satake’s results are valid so the image of H is the set of elements in the group algebra which are invariant under the Weyl group. Suppose φ is bi-invariant under K and

λ(f)φ(g) = ∫_{G_{K}} f(h^{−1})φ(hg) dg ≡ χ(f)φ(g)

where f ↦ χ(f) is a homomorphism of H into ℂ. One need only study the asymptotic behaviour of φ on the ‘positive Weyl chamber’ in A_{K}. (To avoid inessential complications assume that Z_{K} = A_{K}Z_{O}.) If ψ_{1} and ψ_{2} are two functions on A_{K}/A_{O} then ψ_{1} and ψ_{2} will be said to be equivalent if there is a constant c > 0 such that ψ_{1}(a) = ψ_{2}(a) whenever ⟨α, a⟩ ≥ c for all simple roots α. Let W be the vector space formed by these equivalence classes. The group algebra of A_{K}/A_{O} acts on W. Let ψ(a) = L(a, −ρ)φ(a) and let Ψ be the class of ψ in W. Once one shows that Ψ is annihilated by an ideal of finite codimension in the group algebra one sees that there is a closed expression for ψ which is valid if ⟨α, a⟩ is sufficiently large for all simple roots. It is enough to show that

λ(f̃)Ψ = χ(f)Ψ

if f is in H. Choose a compact set ω in U_{K} such that the support of f is contained in ωU_{K}K. If a^{−1}ωa ⊆ U_{O} then

χ(f)φ(a) = ∫_{K} ∫_{A_{K}} ∫_{U_{K}} f(u^{−1}b^{−1}k^{−1})φ(kbua)L(b, −2ρ) du db dk
= ∫_{A_{K}} ∫_{ω} f(u^{−1}b)φ(kba)L(b, −2ρ) du db
= ∫_{A_{K}} φ(ba) ∫ {∫_{U_{K}} f(b^{−1}u) du} db
= ∫_{A_{K}} L(b, −ρ)φ(ba)f̃(b^{−1}) db .

Of course the most interesting feature of Jacquet’s thesis is the functional equation. As I mentioned to you before I believe there is a uniqueness theorem behind it. Consider first the case that K = ℝ. For simplicity take G_{ℝ} connected. Let χ be a generic character of U_{K} and let L(χ) be the space of all K-finite (there are two different K’s here) functions on G_{ℝ} satisfying

(i) φ(gu) = \overline{χ(u)}φ(g) for all u in U_{K} (ii) There are vectors λ_{1}, …, λ_{n} and constants a_{1}, …, a_{n} such that

|φ(ka)| = ∑_{i=1}^{n} a_{i}|⟨λ_{i}, a⟩| .

Do most, if not all, quasi-simple irreducible representations of the universal enveloping algebra occur at most once in L(χ)? I have no idea at the moment how to prove such a uniqueness theorem when the rank is greater than one. Such a proof would be of use for studying similar questions in the harmonic analysis on G and Γ\G. However in the rank-one case it might be possible to prove it by using an analogue of what Harish-Chandra calls the Maass-Selberg relations.

Suppose then the rank of G is one. Let KAN be the Iwasawa decomposition and let M be the centralizer of A in K. Let ℨ be the centre of the universal enveloping algebra 𝔄 of G and ℨ_{1} the centre of the universal enveloping algebra of M. The normalizer of A acts on the homomorphisms of ℨ_{1} into ℂ. Let ζ_{1}, …, ζ_{ℓ} be the orbit of the homomorphism ζ and let L(s, ζ) be the space of K-finite functions on G_{ℝ}/U_{ℝ} which satisfy (i) φ(ga) =< a, −sα − ρ > φ(g) for a ∈ A_{ℝ} (α is the simple root) (ii) ∏_{i=1}^{ℓ} (ρ(Z) − ζ_{i}(Z)) = 0 for Z ∈ Z_{1}. Let π_{s,ζ} be the representation of 𝔄 on L(s, ζ). All representations of 𝔄 which are relevant to us are contained in some π_{s,ζ}. If Z ∈ ℨ then π_{s,ζ}(Z) is a scalar.

Fix an irreducible representation π of K. Let L(χ, s, ζ, π) be the set of all functions in L(χ) which transform under K according to π and satisfy

(A)  λ(Z)φ = π_{s,ζ}(Z)φ .

For such φ there is a constant vector Ψ and a vector function Φ(t) defined for t > 0 such that

φ(ka) = ^{t}Φ(t)π(k^{−1})Ψ

if ⟨a, α⟩ = t^{−1}, α being the simple root of A. Let D = td/dt. Using Harish-Chandra’s results on the structure of the centre of the universal enveloping algebra one should be able to show that equation (A) is equivalent to an equation of the form

∑_{k=0}^{n(Z)} A_{k}(t, Z)D^{k}Φ = π_{s,ζ}(Z)Φ

where A_{k}(t, Z) is a polynomial in t. If one replaces χ by the trivial character and repeats the process one obtains the equation

∑_{k=0}^{n(Z)} A_{k}(0, Z)D^{k}Φ = π_{s,ζ}(Z)Φ .

Moreover if D is the Casimir operator then n(D) = 2 and A_{2}(0, D) is a non-zero scalar. Thus one can presumably apply the method of Frobenius to show that, except for certain exceptional values of s,

Φ(t) = t^{+s+a_{0}}(Φ_{1} + O(t)) + t^{−s+a_{0}}(Φ_{2}(t) + O(t))

if ρ = a_{0}α.

One should also be able to show by using the methods of the theory of Eisenstein series or of asymptotic expansions of ordinary differential equations that if M is real then Φ(t) = O(t^{M}) as t → ∞. Suppose φ lies in L(χ, s, ζ, π) and φ′ lies in L(χ̃, s′, ζ′, π̃). π̃ is the contragredient of π and χ̃ is the contragredient of χ. If B_{x} is the image in N\G of {kan | ⟨a, α⟩ < −x} then

{π_{sζ}(D) − π_{s′,ζ′}(D)} ∫_{B_{x}}{λ(D)φφ′ − φλ(D)φ′} dḡ

is equal to

const × t^{−2a_{0}} \{^{t} (t \frac{d}{dt}Φ(t)) Φ′(t) −^{t}Φ(t) (\overline{t\frac{d}{dt}Φ′(t)})\}_{t=x} ×^{t}ΨΨ̄′ .

In particular if s = s′, ζ = ζ′ the expression

t^{−2a_{0}} \{^{t} (t\frac{d}{dt}Φ(t)) Φ′ −^{t}Φ(t) (t\frac{d}{dt}Φ′(t))\}

must vanish identically. For general values of s this should imply that

(s + a)Φ_{1}Φ′_{1} + (−s + a)Φ_{2}Φ′_{2} − (−s + a)Φ_{1}Φ′_{2} − (s + a)Φ_{2}Φ′_{1} = 0

or

2s{Φ_{1}Φ′_{2} − Φ_{2}Φ′_{1}} = 0 .

This appears to be the analogue of the relation on page 170 of Maass’s paper on non-analytic automorphic forms. One should be able to use it to prove uniqueness theorems.

If φ lies in L(χ, s, ζ, π) and φ′ lies in L(χ, s′, ζ′, π) one should also be able to show that

\{π_{s,ζ}(D) − \overline{π_{s′,ζ′}(D)}\} ∫_{B_{x}}\{λ(D)φφ̄′ − φ\overline{λ(D)φ′}\} dg

is equal to

(B)  const × t^{−2a_{0}} \{^{t}(t\frac{d}{dt}Φ(t))(\overline{Φ}′(t)) −^{t}Φ(t) (\overline{t\frac{d}{dt}Φ′(t)})\}_{t=x} ×^{t}ΨΨ̄′ .

Let w be an element of the normalizer of A which takes P to its opposite. Let θ(g, s) be a function in L(s, ζ) of the form

θ(g, s) =< a, −sα − ρ > ^{t}Θπ(k^{−1})Ξ

where Θ lies in L(ζ, π). That is

∏_{i=1}^{ℓ}(π(Z) − \overline{ζ_{i}(Z)})θ = 0, Z ∈ ℨ_{1} .

The integral

∫_{U_{K}} θ(guw, s)χ̄(u) du = φ(g, s)

is the integral of Jacquet. Thus it should converge for Re(s) > 0. Introduce Φ(t, s) as above; then

Φ(t, s) = t^{s+a_{0}}(M_{1}(s)Θ + O(t)) + t^{−s+a_{0}}(M_{2}(s)Θ + O(t)) .

For each s, M_{1}(s) and M_{2}(s) are linear transformations of L(s, π) to itself. If < a, α >= t^{−1} and θ(g, s) =< a, −sα − ρ > π̄(k)Θ then

\begin{aligned}Φ(ts) & = ∫_{U_{K}} θ(auw, s)χ̄(u) du \\ & = ∫_{U_{K}} θ(aua^{−1}w(w^{−1}aw), s)χ̄(u) du . \end{aligned}

Since w^{−1}aw = a^{−1} this is equal to

⟨a, sα − ρ⟩ ∫_{U_{K}} θ(uw, s)χ̄(a^{−1}ua) du .

Now ⟨a, sα − ρ⟩ = t^{−s+a_{0}} and, as t → 0, a^{−1}ua → 1. It follows that if Re(s) > 0

M_{2}(s)Θ = ∫_{U_{K}} θ(uw, s) du .

If π is the trivial representation of K, Harish-Chandra has shown in his paper on spherical functions that M_{2}(s) can be analytically continued in the whole plane as a meromorphic function. Although he has not published all the details, he can I believe handle the general case. Anyhow assume for the sake of the argument that M_{2}(s) can be analytically continued. By examining the behaviour of M_{2}(s) as s → ∞ one can probably show that detM_{2}(s) ≢ 0 so that M_{2}(s)^{−1} is also meromorphic.

Assume also that the method of Frobenius allows one to show that

Φ(t, s) = t^{s+a_{0}}N_{1}(t, s)M_{1}(s)Θ + t^{−s+a_{0}}N_{2}(t, s)M_{2}(s)Θ

where

N_{i}(t, s) = I + O(t)

uniformly for s in a compact set. (Of course one will probably have to avoid certain exceptional s.) Let

t\frac{d}{dt}N_{i}(t, s) = D_{i}(t, s) .

Since π_{s,ζ}(D) = cs^{2} + d where d is real and c is positive one can, if s is not real or pure imaginary, take φ(g) = φ(g, s) and φ′(g) = φ′(g, s) in formula (B) to see that

− 1/(s^{2} − s̄^{2}) times
t[{(s + a_{0})t^{s}N_{1}(t, s)M_{1}(s) + t^{s}P_{1}(t, s)M_{1}(s) + (−s + a_{0})t^{−s}N_{2}(t, s)M_{2}(s) + t^{−s}P_{2}(t, s)M_{2}(s)}Θ]
k×\overline{[\{t^{s}N_{1}(t, s)M_{1}(s) + t^{−s}N_{2}(t, s)M_{2}(s)\}Θ′]}
−^{t}[{t^{s}N_{1}(t, s)M_{1}(s) + t^{−s}N_{2}(t, s)M_{2}(s)}Θ]
×\overline{[\{(s+a_{0})t^{s}N_{1}(t, s)M_{1}(s) + t^{s}P_{1}(t, s)M_{1}(s)+(−s+a_{0})t^{−s}N_{2}(t, s)M_{2}(s) + t^{−s}P_{2}(t, s)M_{2}(s)\}Θ′]}

is a positive semi-definite hermitian form on Θ and Θ′. This expression equals, if s = σ + iτ,

\frac{1}{2σ} [t^{−2σ}(N_{2}(t, s)M_{2}(s)Θ, N_{2}(t, s)M_{2}(s)Θ′) − t^{2σ}(N_{1}(t, s)M_{1}(s)Θ, N_{1}(t, s)M_{1}(s)Θ′)]

minus

\frac{1}{2iτ} [t^{2iτ} (N_{1}(t, s)M_{1}(s)Θ, N_{2}(t, s)M_{2}(s)Θ′) − t^{−2iτ} (N_{2}(t, s)M_{s}Θ, N_{1}(t, s)M_{1}(s)Θ′)]

minus four other terms involving P_{i}(t, s) of which one would be

t^{2σ} [^{t}Θ^{t}M_{1}(s) \{\frac{^{t}P_{1}(t, s)N̄_{1}(t, s) −^{t}N_{1}(t, s)P̄_{1}(t, s)}{s^{2} − s̄^{2}}\} M̄_{1}(s)Θ̄′] .

If the last four terms were not present this expression would be just like the one used to effect the analytic continuation of Eisenstein series. Since the last four terms are O(t) it is not inconceivable that the same techniques could be used to handle Jacquet’s integral.

It is going to be more difficult to handle the continuation when K is non-archimedean and, at the moment, I have no suggestions. It seems to me however that when G is quasi-split, split over an unramified extension, D is trivial, and the character ζ is generic it should be possible to obtain a simple closed expression for what Jacquet calls the Whittaker function. The only case I have thought about is the adjoint group of a Chevalley group over ℚ_{p}.

Let w be an element of the Weyl group which takes all positive roots to negative roots. Suppose ζ is such that the μ_{α} on page 44 of Jacquet are all 1. Rather than the Whittaker function consider

θ(g, λ) = ∫_{U_{K}} L(guw, λ + ρ)\overline{ζ(u)} du .

Certainly θ(g, λ) satisfies

(i) θ(gu, λ) = ζ(u)θ(g, λ) u ∈ U_{K} (ii) θ(mg, λ) = θ(g, λ) m ∈ M Any function which satisfies (i) and (ii) is determined by its restriction to A_{K}. Moreover if u ∈ U_{K} ∩ M then

θ(a, λ) = θ(ua, λ) = ζ(a^{−1}ua)θ(a, λ) .

Consequently θ(a, λ) = 0 unless ⟨a, α⟩ ≥ 1 for all positive roots so that a lies in the ‘positive Weyl chamber’. By the way using this one should be able to show by an inductive argument that if a function satisfies (i) and (ii), is an eigenfunction of the Hecke algebra, and vanishes at 1 then it vanishes identically. This would be a simple uniqueness theorem.

Set

ω(g, λ) = ∏_{α>0} p^{λ(H_{α})/2} \{\frac{1 − \frac{1}{p^{λ(H_{α})}}}{1 − \frac{1}{p^{λ(H_{α})+1}}} \} θ(g, λ) .

The functional equation of Jacquet is just

ω(σλ, g) = sgn σ ω(λ, g)

if σ is in the Weyl group. (He does not work with the adjoint group but that is no matter.) The function ω(g, λ) appears to be an entire function of λ. Thus it can be expanded in a Fourier series. The resulting formula will be more poignant if one makes a simple observation first. Consider the following objects.

{α̂_{1} = H_{α_{i}}|α_{i} a simple root} {α_{i} = Hα̂_{i}|α_{i} a simple root}
⋂|  ⋂|
{α̂ = H_{α}|α a root}  {α = Hα̂|α a root}
⋂|  ⋂|
Hom (L, ℤ) = L̂′  L′ = lattice spanned by roots
⋂|  ⋂|
Hom (L′, ℤ) = L̂  L integral linear functions
⋂|  ⋂|
Cartan subalgebra of g_{ℚ} = h_{ℚ} ĥ_{ℚ} = dual of h_{ℚ}
⋂|  ⋂|
h_{ℝ} ĥ_{ℝ}
⋂|  ⋂|
h_{ℂ}  ĥ_{ℂ}

There is another semi-simple group Ĝ over ℚ with a Cartan subalgebra which may be identified with ĥ_{ℚ} so that the roots correspond to {α̂}. Then there is a duality between the two columns. Moreover there is an isomorphism a → λ̂(a) of A_{K}/A_{O} with L̂ which is such that ⟨a, λ⟩ = p^{⟨λ,_{λ}(a)⟩} if ⟨λ, λ̂⟩ is the pairing of h_{ℚ} and ĥ_{ℚ}. Then

ω(a, λ) = \{ ∏_{α>0} \frac{p^{⟨λ,_{α}⟩/2} − p^{−⟨λ,_{α}⟩/2}}{1 − \frac{1}{p^{⟨λ,_{α}⟩+1}}} \} θ(a, λ)

and

ω(a, λ) = ∑_{_{λ}∈_{L}} γ′(a, λ̂)p^{⟨λ,_{λ}⟩}

Because of the skew symmetry the right side may be written as

∑_{_{λ}∈_{L^{+}}} γ(a, λ̂) ∑_{_{σ}∈ Weyl group} sgn σ̂p^{⟨λ,_{σ}(_{λ}+_{ρ})⟩}

if L̂^{+} is the intersection of L̂ with the positive Weyl chamber. It seems to be the case that if n ∈ U_{K} and m_{1}a_{1}n_{1} is the Iwasawa decomposition of nw then ⟨λ̂(a_{1}), α⟩ ≥ 0 if α > 0 and λ̂(a_{1}) = 0 if and only if n ∈ M. It follows that γ(a, λ̂) = 0 unless ⟨α, λ̂(a) − λ̂⟩ ≥ 0 for all α > 0 and that

γ(a, λ̂(a)) = p^{−⟨ρ,_{λ}(a)⟩} .

I would not be surprised if γ(a, λ̂) were 0 for λ̂ ≠ λ̂(a).

For your amusement I would like to mention a possible application of the ideas of Jacquet’s thesis. Unfortunately it will not work until his results are improved slightly. Suppose G^{0} is the adjoint group of a Chevalley group over ℚ and suppose ϕ is a cusp form on G_{𝔸}^{0}/G_{ℚ}^{0} which is invariant under M and is an eigenfunction of the Hecke algebra H_{p} at every prime including infinity. Let χ_{p} be the homomorphism of the Hecke algebra into ℂ associated to p. If p is finite then H_{p} is isomorphic to the representation ring of the simply connected group Ĝ^{0}. Thus there is a semi-simple conjugacy class {g_{p}} in Ĝ_{ℂ}^{0} such that if f in H_{p} corresponds to the representation π then χ_{p}(f) = traceπ(g_{p}). To χ_{∞} one can associate a semi-simple conjugacy class in the Lie algebra 𝔤̂_{ℂ}^{0}. If π is a representation of G^{0} choose X_{∞} in diagonal form and set

Γ(s, π, ϕ) = ∏_{_{λ}∈_{L}} \{π^{−(s−_{λ}(X_{∞})/2}Γ (\frac{s − λ̂(X_{∞})}{2})\}^{m_{λ}}

where m_{λ} is the multiplicity with which λ̂ occurs in π. If

ξ(s, π, ϕ) = Γ(s, π, ϕ) ∏_{p finite} \frac{1}{det (I − \frac{π(g_{p})}{p^{s}})}

then ξ(s, π, ϕ) is analytic in a half-plane. Beyond that nothing is known for general π. Let me show you how the ideas of Jacquet might be applied to obtain a functional equation for particular choices of ϕ and π.

Suppose G is also the adjoint group of a Chevalley group over ℚ, P ⊇ B a parabolic subgroup of rank one, and P = ZN the decomposition of P as the product of a reductive Z and the unipotent radical N. Let C be the centre of Z and suppose that G^{0} ≅ Z/C ≅ P/NC. Because of the map P_{𝔸} → G_{𝔸}^{0} one can regard ϕ as a function on P_{𝔸}. If p ∈ P let χ(p) be the determinant of Ad(p) acting on 𝔫, the Lie algebra of N, and set

F (g, s, ϕ) = |χ(b)|^{−s−1/2} ϕ(b)

if g = mb, m ∈ M, b ∈ P_{𝔸}. The Eisenstein series is

E(g, s, ϕ) = ∑_{G_{ℚ}/P_{ℚ}} F (gγ, s, ϕ) .

Let ′P ⊇ B be conjugate to the parabolic group opposed to P. Let ω_{0} in the normalizer of A in M be such that ω_{0}′P ω_{0}^{−1} is opposed to P. Let Z′ = ω_{0}^{−1}Zω_{0}, C′ = ω_{0}^{−1}Cω_{0}, and set ′G^{0} = Z′/C′. Let ϕ′ be he function on ′G^{0} defined by

ϕ′(m′) = ϕ(ω_{0}m′ω_{0}^{−1}) .

The functional equation connects E(g, s, ϕ) and E(g, −s, ϕ′). To describe it complete the two columns on page 18 by

G Ĝ - simply connected
∪  ∪
P P̂
↓  ↓
G^{0} Ĝ^{0} - simply connected

Let 𝔫̂ be spanned by {X_{α} | α̂ ∈ θ} and set

H_{0} = ∑_{_{α}∈θ} H_{α} .

Let a_{1}, …, a_{r} be the eigenvalues of H_{0} acting on 𝔫̂ and let 𝔫̂_{1}, …, 𝔫̂_{r} be the corresponding subspaces. Let π_{i} be the representation of Ĝ^{0} on 𝔫̂_{i} and π̃_{i} its contragredient. Then

E(g, s, ϕ) = \{∏_{i=1}^{r} \frac{ξ(a_{i}s, π̃_{i}, ϕ)}{ξ(a_{i}s + 1, π̃_{i}, ϕ)}\} E(g, −s, ϕ′) .

Using this one can show, at least for simple groups, that ξ(s, π_{i}, ϕ) and ξ(s, π̃_{i}, ϕ) are meromorphic in the whole plane.

Suppose ξ is a character of U_{𝔸}/U_{ℚ} so that the μ_{α} on page 93 of Jacquet are all 1. Set

η(g, s, ϕ) = ∫_{U_{𝔸}/U_{ℚ}} E(gu, s, ϕ)ξ̄(u) du .

What I want to point out is that the results of Jacquet are almost good enough to show that

\begin{aligned}η(g, s, ϕ) & = \{∏_{i=1}^{r} \frac{1}{ξ(a_{i}s + 1, π̃_{i}, ϕ)}\} μ(g, s, ϕ) \\ η(g, s, ϕ′) & = \{∏_{i=1}^{r} \frac{1}{ξ(a_{i}s + 1, π_{i}, ϕ)}\} μ′(g, s, ϕ′) \end{aligned}

with μ(g, s, ϕ) = μ′(g, −s, ϕ′). For many ϕ the function μ(g, s, ϕ) will not vanish identically. Then one deduces that

∏_{i=1}^{r} ξ(a_{i}s, π̃_{i}, ϕ) = ∏_{i=1}^{r} ξ(1 − a_{i}s, π_{i}, ϕ)

and then, at least for simple groups,

ξ(s, π̃_{i}, ϕ) = ξ(1 − s, π_{i}, ϕ) .

I leave the calculations to Jacquet if he is interested. The biggest obstacle will be a uniqueness theorem at the infinite prime.

I apologize for the technicality of this letter. I felt that the best way I had of indicating the interest of Jacquet’s ideas would be to explain, as well as I could, their possible implications for the theory of group representations and automorphic forms. Unfortunately, although I had been thinking about the implications since I first noticed Jacquet’s papers I had written nothing down. Because I had to hurry all my suggestions had to be tentative. I hope there is something in them of value which has not occurred to you or Jacquet. Yours truly, R. Langlands Let me add some remarks not related to Jacquet’s thesis. Since you were rather skeptical about the interest of the functions ξ(s, π, ϕ) when I spoke to you in Princeton I would like to comment on their relation to a generalized Ramanujan conjecture. For the ordinary Ramanujan conjecture one has to consider functions which are not M invariant. I have not yet tried to understand the complications this entails. Let Û^{0} be a maximal compact subgroup of Ĝ_{ℂ}^{0}. The generalized Ramanujan conjecture for the function ϕ on p. 20 would say that when p is finite the conjugacy class {g_{ρ}} meets Û^{0} and that the conjugacy class {X_{∞}} meets the Lie algebra of Û^{0}. It implies that Γ(s, π, ϕ) is analytic for Re x > 0 and that the Euler product on p. 20 converges for Re s > 1 so that ξ(s, π, ϕ) is analytic for Re s > 1.

Conversely suppose that, for all π, ξ(s, π, ϕ) is analytic for Re s > 1. If p is finite then H_{1} is isomorphic to the representation ring of Ĝ over ℂ. The involution f → f̃ with f̃(g) = f̄(g^{−1}) corresponds to the involution Σa_{ρ}ρ → Σ\overline{a}_{ρ}ρ̃. Since χ_{ρ}(f̃) = \overline{χ}_{ρ}(f) one has trace ρ̃(g_{ρ}) = \overline{trace ρ(g_{ρ})}. In the same way the eigenvalues of ρ̃(X) are the complex conjugates of those of ρ(X). Take a representation ρ and let π = ρ ⊗ ρ̃. Since ξ(s, π, ϕ) is analytic for Re s > 1 and the Γ-function has no zeros so is

L(s, π, ϕ) = Π_{p finite} \frac{1}{det (L − \frac{π(g_{p})}{p^{s}})}.

Observe that the coefficients of the Dirichlet series log L(s, π, ϕ) and hence those of L(s, π, ϕ) are positive because

log L(s, π, ϕ) = Σ_{1}Σ_{n=1}^{α} \frac{trace π^{n}(g_{ρ})}{ρ^{ns}}

and

trace π^{n}(g_{ρ}) = trace ρ^{n}(g_{0}) · trace ρ̃^{n}(g_{ρ}) = | trace ρ^{n}(g_{0})|^{2}

By Landau’s theorem the series converges absolutely for Re s > 1. In particular

det (1 − \frac{π(g_{ρ})}{ρ^{s}})

does not vanish for Re s > 1 so that the eigenvalues of π(g_{ρ}) are all less than or equal to ρ in absolute value. Choose g_{ρ} in the Cartan subgroup Â_{ℂ}^{0} and let λ̂ be a weight. Given an integer m′ choose ρ so that mλ̂ occurs in ρ. Then (ξ_{λ}(g_{ρ}))^{m} is an eigenvalue of ρ(g_{ρ}) so \overline{(ξ_{λ}(g_{ρ}))^{n}} is one eigenvalue of ρ̃(g_{ρ}) and |ξ_{λ}(g_{ρ})|^{2m} is an eigenvalue of π(g_{ρ}). Consequently

p^{−1/2m} = |ξ_{λ}(g_{ρ})| ≤ p^{1/2m}

for all m > 0. This takes care of the finite primes.

Choose X_{∞} in the Cartan subalgebra. Since L(s, π, ϕ) cannot vanish for Re s > 1, Γ(s, π, ϕ) will be analytic in this region. From the expression for Γ(s, π, ϕ) one concludes that all the eigenvalues of π(X_{∞}) have real part at most 1. Again given λ̂ choose ρ so that in λ̂ is a weight of ρ. Then 2mRe (λ̂(X_{∞})) is an eigenvalue of π. Thus

−\frac{1}{2m} ≤ Re λ̂(X_{∞}) ≤ \frac{1}{2m}

for all m > 0. This takes care of the infinite prime.

The series also seem to be related to Sato’s conjecture. I too have been thinking a little about Weil’s paper. I plan to spend the next month getting my thoughts organized. It will take me a while to digest the second part of your letter. Do you still plan to come to Princeton for 1968-1969? Unfortunately Hunt put me in such a position that I had no choice but to resign from Princeton. I will be going to Yale when I return from Ankara so I shall not see much of you if you come. As you probably know I received an invitation from Lions to spend a month in Paris either next year or the year after. I would like to ask your advice. I gave some lectures (I will send you the notes) at Yale last spring on the relation of the functions ξ(s, π, ϕ) to Eisenstein series. I would like, for my own information, to work things out for general automorphic forms on general reductive groups. After that I want to try to formulate exactly the definition of ξ(s, π, ϕ) in the general case. Then I have to try to prove something. Since I don’t have the foggiest idea how to proceed at the moment it might be a while (never?) before I get anything interesting. Since I would be expected to give some lectures in Paris I should have something to say. Do you think that I could write to Lions and say that I should like very much to come but do not know when I could be prepared; that I would tell him when I was prepared and then if he still cared to he could renew the invitation. It would also be best for me to come when you are there.

Yours truly, R. Langlands