Representations of the group \(\text{GL}(n,F),\) where \(F\) is a local non-archimedean field. (Russian) Zbl 0342.43017

Bernshteĭn, I. N.; Zelevinskiĭ, A. V.

Usp. Mat. Nauk 31, No. 3(189), 5-70 (1976).

This article is a survey of recent results on representations of reductive \(p\)-adic groups. The main model is the group \(\text{GL}(n, F)\), where \(F\) is a local non-archimedean field. Chapter I: Preliminaries. Chapter II: the theory of Harish-Chandra concerning the representation of \(\text{GL}(n, F)\) is presented. New proofs of theorems of R. Howe on the finiteness of an admissible representation of \(\text{GL}(n, F)\) are given. Chapter III is devoted to the theory of to Gel’fand and Kazhdan. The main idea of their methods lies in the fact that the restriction of an irreducible representation of \(\text{GL}(n, F)\) to a certain subgroup is almost equivalent to a standard irreducible representation of this subgroup. A generalization of a theorem of Jacquet-Langlands on the codimension of a finite function in the Kirillov model is proved. All theorems are given with complete proofs.

Reviewer: E. Plonka

MSC:

22E50

Representations of Lie and linear algebraic groups over local fields