New Haven September 22, 1969

Dear Harish-Chandra,

I am very pleased that you are willing to come here to speak. I have decided to come to Princeton on Monday, October 7 and to stay over till Tuesday. However I don’t want to give a lecture then because Weil and Jacquet are familiar with anything conclusive I did before January and I have not yet finished the things I have been doing since. I have already given you an idea what these things are but I can perhaps be more precise now.

If F is a local field let C_{F} be the multiplicative group of F and if F is a global field let C_{F} be the idèle class group of F. As I said before if K\F is normal the Weil group G_{K|F} is an extension of C_{F} by the Galois group of K\F. If one likes one can take projective limits and get an object called the Weil group of F. If, F is global, and F_{𝔭} is a completion of F, and K_{𝔓} is a completion of K over 𝔭 then there is a map of G_{K_{𝔓}}\F_{𝔓} into G_{K_{𝔓}}\F_{𝔭}. A representation ρ is a finite-dimensional representation over ℂ such that ρ(g) is semi-simple for all g.

For a non-archimedean local field I can attach to every equivalence class ω of representations a function L(s, ω) in just one way so that the following conditions are satisfied.

(i) If ω ∼ χ_{F} is one dimensional

L(s, ω) = \frac{1}{1−χ_{F} (π_{F})|π_{F} |^{s}} if χ_{F} is trivial on units
= 1  if χ_{F} is not trivial on units.

(ii) L(s, ω_{1} ⊕ ω_{1}) = L(s, ω_{1})L(s, ω_{2}). (iii) If ω_{1} ≃ Ind (G_{K\F} , G_{K\E}, ω_{2}) then

L(s, ω_{1}) = L(s, ω_{2}).

For archimedean fields L(s, ω) is defined by the following conditions

(i) If F = ℝ and ω ∼ χ_{ℝ} with χ_{ℝ}(x) = (sgn x)^{m}|x|^{r}, m = 0 or 1, then

L(s, ω) = π^{− \frac{1}{2} (s+r+m)}Γ (\frac{s + r + m}{2}) .

(ii) If F = ℂ and ω ∼ χ_{ℂ} with χ_{ℂ}(z) = |z|^{2r} \frac{z^{n}z̄^{n}}{|z|^{m+n}} , m + n ≥ 0, mn = 0 then

L(s, ω) = 2(2π)^{−(s+r+ \frac{m+n}{2} )}Γ(s + r + \frac{m + n}{2})

(iii) L(s, ω_{1} ⊕ ω_{2}) = L(s, ω_{1})L(s, ω_{2}) (iv) If ω_{1} ≃ Ind (G_{K\F} , G_{K\E}, ω_{2}) then

L(s, ω_{1}) = L(s, ω_{2})_{.}

If K is a global field, ω is an equivalence class of representations of the Weil group and ω_{𝔭} the induced equivalence class of the Weil group of F_{𝔭} the Artin L-function is

L(s, ω) = Π_{𝔭}L(s, ω_{𝔭})

Let me state the theorem which I have been working on since last January and then comment on its relation to the functional equations of the L-functions. By the way, as soon as I understand a lemma of Dwork I should be able to write out the proof of this theorem. I think I have all the parts of the proof except for some easily managed details. If F = ℝ and χ_{ℝ}(x) = (sgn x)^{m}|x|^{r} with m = 0 or 1 and ψ_{ℝ} is the additive character ψ_{ℝ}(x) = e^{2πiux} I set

Δ(χ_{ℝ}, ψ_{ℝ}) = (i sgn u)^{m}|u|^{r}.

If F = ℂ and χ_{ℂ}(z) = |z|^{2r} \frac{z^{m}z̄^{n}}{|z|^{m+n}} , m+n ≥ 0, mn = 0 and ψ_{ℂ} is the additive character ψ_{ℂ}(z) = e^{4πiℜ(wz)}. I set

Δ(χ_{ℂ}, ψ_{ℂ}) = i^{m+n}χ_{ℂ}(w).

If F is non-archimedean, if χ_{F} is a generalized character of C_{F} with conductor 𝔓_{F}^{m}, if ψ_{F} is a non-trivial additive character of F and 𝔓_{F}^{−n} is the largest ideal on which it vanishes and O_{F} γ = 𝔓_{F}^{m+n} I set

Δ(χ_{F} , ψ_{F}) = χ_{F} (γ)\frac{∫_{U_{F}} ψ_{F} (\frac{α}{γ})χ_{F}^{−1}(α)dα}{| ∫_{U_{F}} ψ_{F} (\frac{α}{γ})χ_{F}^{−1}(α)dα|}.

U_{F} is the group of units. The right side is independent of γ.

If E is a separable extension of F and ψ_{F} is given then

ψ_{E\F} (X) = ψ_{F} (T_{r_{E\F}} X).

Theorem. Suppose F is a given local field and ψ_{F} a given non-trivial additive character of F. It is possible in exactly one way to assign to each separable extensions E of F a complex number ρ(E | F, ψ_{γ}) and to each equivalence classic of representations of the Weil group of E a complex number ϵ(ω, ψ_{E|F}) so that

(i) If ω ≃ χ_{E} then ϵ(ω, ψ_{E|F}) = Δ(χ_{E}, ψ_{E|F}) (ii) ϵ(ω_{1} ⊕ ω_{1}, ψ_{E|F}) = ϵ(ω_{1}, ψ_{E|F})ϵ(ω_{2}, ψ_{E|F})

(iii) If ω_{1} ≃ Ind (G_{K\F} , G_{K\E}, ω_{2}) then

ϵ(ω_{1}, ψ_{F}) = ρ(E | F, ψ_{F})^{dim ω_{2}}ϵ(ω_{2}, ψ_{E|F}).

If A_{F}^{s} is the generalized character α → |α|_{F}^{s} set ϵ(s, ω, ψ_{F}) = ϵ(A_{F}^{s−1/2} ⊗ ω, ψ_{F}). If F is a global field and ψ_{F} a non-trivial character of 𝔸_{F} /F let ψ_{F_{𝔭}} be the restriction of ψ_{F} to F_{𝔭}. If ω is an equivalence class of representations of the Weil group of F and

ϵ(s, ω) = Π_{𝔭}ϵ(s, ω_{𝔭}, ψ_{F_{𝔭}})

the functional equation of the L-function can, on the basis of the previous theorem, be shown to be

L(s, ω) = ϵ(s, ω)L(1 − s, ω̃)

if ω̃ is contragredient to ω.

Once I have this theorem I should very quickly be able to derive a consequence of interest for group representations. However I have not yet carried out the computations. After some preliminaries I shall state the consequence.

If F is a non-archimedean local field a two-dimensional equivalence class ω of representations of the Weil group of F will be called special if ω is the direct sum of two one-dimensional representations μ_{F} and ν_{F} and μ_{F} ν_{F}^{−1} = A_{F}^{1} or A_{F}^{−1}. If F = ℝ a two-dimensional equivalence class ω is special if ω ≃ μ_{F} ⊕ ν_{F}, and μ_{F} (x) = |x|^{s_{1}} (\frac{x}{|x|})^{m_{1}} , ν_{F} (x) = |x|^{s_{1}} (\frac{x}{|x|})^{m_{2}} and (s_{1} − s_{2}) − (m_{1} − m_{2}) is an odd integer. If F = ℂ, ω is special if ω ≃ μ_{F} ⊕ ν_{F}

\begin{aligned}μ_{F} (z) & = |z|^{2s_{1}} (\frac{z}{|z|})^{m_{1}} \\ ν_{F} (z) & = |z|^{2s_{2}} (\frac{z}{|z|})^{m_{2}} \end{aligned}

and one of \frac{s_{1}−s_{2}}{2} − (1 + \frac{|m_{1}−m_{2}|}{2}) and \frac{s_{2}−s_{1}}{2} − (1 + \frac{|m_{1}−m_{2}|}{2}) is a non-negative integer. L(ψ_{F}) is the space of functions on GL(2, F) satisfying

φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) ≡ ψ_{F} (x)φ(g)

together with certain conditions on smoothness and rate of growth. Here is the consequence I mentioned.

Theorem. Suppose that for every global field F and every two-dimensional irreducible equivalence class of representations of the local group of F the function L(s, ω) is entire and bounded in vertical strips. Then if F is a local field, ω a two-dimensional equivalence class of representations of the Weil group of F which is not special, and ψ_{F} a non-trivial additive character of F there is a unique simple representation π_{ω} of GL(2, F) satisfying

(i) π_{ω} acts on L ⊆ L(ψ_{F}) (ii) If φ belongs to L and χ_{F} is a generalized character of C_{F} the integral

∫_{C_{F}} φ (( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) g) χ_{F} (α)|α|_{F}^{s} dα

converges for Re s sufficiently large. Denote its value by Φ(g, s, φ, χ_{F}) and set

Φ(g, s, φ, χ_{F}) = L(s + \frac{1}{2}, ω ⊗ χ_{F})Φ′(g, s, φ, χ_{F})

Φ′(g, s, φ, χ_{F}) is an entire function of s bounded in vertical strips. Moreover

Φ′ (( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) g; −s, φ(detωχ_{F})^{−1}) = ϵ(χ_{F} A_{F}^{s} ⊗ ω, ψ_{F})Φ′(g, s, φ, χ_{F})

if det ω is the 1-dimensional representation obtained from ω by taking determinants.

It will follow that π_{ω_{1}} equivalent to π_{ω_{2}} implies ω_{1} = ω_{2}. This theorem makes the existence of the representations I mentioned to you earlier pretty much certain. If they do not exist the L-series behave in an entirely unexpected manner.

Yours truly,

Bob Langlands