Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$ - MathOverflow

Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$

2012-02-21T11:42:44Z

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 ?

Answer by Marty for Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$

2012-02-21T18:41:47Z

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.