Letter to Godement

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Editorial comments: The letter to Weil that saw the birth of the \(L\)-group was written in January, 1967. Somewhat later that same year, Roger Godement asked Langlands to comment on the Ph. D. thesis of Hervé Jacquet. His reply included a number of conjectures on Whittaker functions for both real and \(p\)-adic reductive groups. These were later to be proven, first in the \(p\)-adic case by Shintani for \(\mathrm{GL}_n\) and Casselman Shalika in general, and much later in the real case by a longer succession of people.

Author's comments: This letter, a report on Jacquet's thesis, is undated, but a letter from Godement dated May 12th, 1967 asks that the report be submitted before the end of May. I assume it was sent from Princeton so as to arrive in Paris before the date requested.

The notation may cause the reader some difficulties. Some symbols, for example \(\chi\), have meanings that change (sometimes explicitly but sometimes only implicitly) in the course of the letter. There is a particularly dangerous lapse in regard to \(\xi\). Other symbols, sometimes the same, are employed in ways that have become uncommon. The symbol \(\pi\) appears, for example, as a representation of a compact group. The notation \(\langle a, \alpha\rangle\) for the value of the multiplicative function \(\alpha\) at the group element \(a\) is particularly disconcerting.

References to pages either in Jacquet's thesis or in the handwritten letter have been allowed to stand.

The formula for Whittaker functions for unramified representations suggested in the letter was proved by Casselman and Shalika.

It appears from the Institute records that Godement visited Princeton early in March of 1967. It must have been then that I spoke to him. The lectures at Yale were given early in April of 1967 and appeared later as the monograph Euler Products (included just above).


Type: 
article
Year: 
1967
School of Mathematics: 
Author: 
Robert P. Langlands