MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question is out of plain curiosity. The first sentence of Deligne's Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as follows :

D. Kazhdan has introduced the principle that the representation theory of a reductive group over a local field of prime characteristic $p$ is the limit, as the ramification index tends to infinity, of theories over local fields of characteristic $0$ with the same residue field.

He says that according to Langlands' philosophy, one should expect the same phenomenon to occur on the galoisian side, and goes on to establish a precise equivalence of categories justifying this principle (and clarifying the earlier work of M. Krasner from the forties).

I'm mainly interested in this side of the story, but I'm curious as to where Kazhdan's principle in representation theory was first enunciated. What are the standard references in English or French explaining this principle ?

share|cite|improve this question
This may be hard to pin down in the published literature, so I'd suggest asking Kazhdan directly. It's helpful anyway to add MathSciNet data for Deligne's paper, where the reviewer E. Zink states that Kazhdan's principle was "recently introduced": MR771673 (86g:11068) Deligne, P. Les corps locaux de caracte ́ristique p, limites de corps locaux de caracte ́ristique 0. (French) [Local fields of characteristic p which are limits of local fields of characteristic 0], Representations of reductive groups over a local field, 119–157, Travaux en Cours, Hermann, Paris, 1984. – Jim Humphreys Feb 21 '12 at 14:33
up vote 14 down vote accepted

I think, although it's dated later than Deligne's paper that you mentioned, that the first written instance of Kazhdan's principle is in the paper "Representations of groups over close local fields", Journal d'Analyse Math\'ematique, vol. 47,1986, pp.175--179.

This is in the same journal issue as "Cuspidal Geometry of p-adic Groups" (by Kazhdan) and "Trace Paley-Wiener Theorem for Reductive p-adic Groups" (by Bernstein, Deligne, Kazhdan). The book "Representations of reductive groups over a local field" appeared in 1984, and according to the MathSciNet review of the article "Le 'Centre' de Bernstein", the Trace Paley-Wiener Theorem paper was already a preprint in 1984.

So it seems to me that Kazhdan's principle was probably "in the air" by 1984, but not written down by him until the "close local fields" article above. I second Jim Humphreys' suggestion to contact Kazhdan himself for less speculative history.

share|cite|improve this answer
@Marty: Your hunch has been confirmed by Professor Kazhdan in an email message. – Chandan Singh Dalawat Feb 22 '12 at 7:04

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.