Letter to Gee

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Author's comments: The Galois representations attached in the context of Shimura varieties to certain automorphic representations usually correspond---under the correspondence of the "Langlands program"---not to the representation to which they are attached but to some twist of it by a central character. There is no reason---logical or aesthetical---that it should be otherwise. Nevertheless a casual search of the literature will probably reveal that a number of authors were troubled by it and attempted---on their own initiative---to revise the definitions. This has led to a certain amount of confusion. My own feeling is that the parametrization at infinity described in "On the Classification of Irreducible Representations of Real Algebraic Groups", depending as it does on curious properties of one-half the sum of positive roots is a kind of miracle, so that it is rash to tamper with it without very, very good reasons.

Of course, the twist, perhaps not so different from that arising in the theory of perverse sheaves, causes, as that does, annoying mnemonic difficulties. Although I have never made the effort, I suppose they can be overcome by thinking the whole matter through carefully.

The letter, written under hurried circumstances in a hotel room, is not so clear as it might be. Indeed, as written it is downright confusing. What is important to underline is that the representation \(\pi'\) is the one with the right properties at infinity, namely it is associated to representations of the local Weil group that are algebraic on the multiplicative group of the complex numbers, a subgroup of each of the local Weil groups. It also, at least for the group of two-by-two matrices, and presumably in general, at the other local places conforms to the local correspondence of the "Langlands program". On the other hand, \(\pi'\) is different from \(\pi\), the representation yielding the complex cohomology on whose \(\ell\)-adic counterpart \(\sigma\) is realized. (The equality \(\pi'_p=\sigma_p\) is nonsense! What I meant was that the Galois representation \(\sigma_p\) and the local component \(\pi'_p\) of the automorphic representation \(\pi\) correspond.) There is, so far as I can see, no harm in this.

I have had occasion since writing this note to look at Clozel's paper. It appears in the first volume of the proceedings of the Ann Arbor conference on Automorphic forms, Shimura varieties and L-functions. Clozel is aware of the difference between \(\pi\) and \(\pi'\) but draws different conclusions from this than I do, at least at the moment. I discuss this and related matters in the comments on the 1974 letter to Deligne.

School of Mathematics: