Ankara April 18, 1968

Dear Harish-Chandra,

I have been wanting to write you for some time to thank you for your letter and to assure you that everything is satisfactory here. However, I had begun to form some suspicions about representations of reductive algebraic groups over local fields and I wanted at least to check these carefully in the case of GL_{2} before writing to you. I see now that I will have left here before I finish so I started to write you anyway and voice the suspicions in a premature form.

I remind you that if k is a local field and K a normal extension then G_{K,k} (the Weil group of K/k) is a certain extension of K^{∗} by the Galois group of K/k. If k ⊆ K ⊆ L with K and L normal there is a homomorphism φ of G_{L,k} into G_{K,k}. This homomorphism is not uniquely determined but if ρ is a representation of G_{K,k} the equivalence class of ρ ∘ φ is uniquely determined. Thus if ρ_{1} and ρ_{2} are representations of G_{K_{1},k} and G_{K_{2},k} respectively I can call them equivalent if they become equivalent when lifted to G_{K_{1}K_{2},k}. Also I can speak of the representations of the local group of k without being explicit about the choice of K. Now in my old letter to Weil I made a rough attempt to define the dual group of a reductive group. The attempt was not satisfactory but provides a basis for thought. I have come to believe that associated to almost every equivalence class of continuous representations of the local group in this dual group, a complex group, which is such that ρ(σ) is semi-simple for all σ there should correspond an equivalence class of representations of the algebraic group. In particular to every unitary representation of the Weil group in this local group should correspond a unitary representation of the algebraic group over k. To give some basis to this belief and to complete the things I was doing with the Jacquet I wanted to check this out completely for GL_{2}. In this case the dual group is GL(2, ℂ).

If the local field is non-archimedean and the characteristic of the residue class field is different from 2 then there are basically only two ways of getting a representation of the Weil group in GL(2, ℂ). Either take K = k so that G_{K,k} = k^{∗} and send α → ( \begin{matrix}χ_{1}(α) & 0 \\ 0 & χ_{2}(α) \end{matrix} ) where χ_{1} and χ_{2} are two characters of k^{∗} or take K to be a quadratic extension of k, take a character χ of K^{∗}, and let χ induce a representation of G_{K,k}. Now give χ_{1}, χ_{2} or χ we know how to construct a representation of GL(2, k). The only thing to check is that if the representations give equivalent representations of the Weil group in the sense mentioned above then the representations of GL(2, k) are equivalent. This I have done, although I should probably look at the proof again. It can, for example, happen that K and K′ are distinct quadratic extensions while the representations induced from χ and χ′ give equivalent representations of the Weil group. Archimedean fields are even simpler.

However, it is very likely that if the characteristic of the residue class field is 2 there are two dimensional representations of the Weil group which are not abelian and are not associated to a character of a quadratic extension of k. This is why I told Jacquet that I considered it unlikely that we had exhausted the representations in this case. Although my work in this is proceeding at a reasonably steady pace it will be a while before it is finished. I hope to be able to tell you something definite when I return in August.

All the best,

Bob Langlands