An external approach to unitary representations. (English) Zbl 0799.22010
In the early 1980’s, the author announced his solution to the problem of classifying the irreducible unitary representations of \(GL_ n(F)\), \(F\) a non-archimedean local field [cf. Ann. Sci. Ec. Norm. Supér., IV. Sér. 19, 335-382 (1986; Zbl 0614.22005)]; he also recognized that the same result holds over \(\mathbb{R}\) and \(\mathbb{C}\) [cf. MPI/SFB 85-22, Bonn (1985)]. The main purpose of the present paper is to present the ideas which lie behind these results. The author stresses that his approach is an “external” one, i.e., no attempt is made to analyze the “internal” structure of the representations involved, for example, their decomposition upon restriction to certain compact subgroups. The first part of the Classification Theorem explicitly describes certain elements of \(\widehat{G}\) (which in the case of \(F = \mathbb{C}\) amounts to the well- known list of Gelfand-Naimark, plus Stein’s “degenerate” complementary series); the second part asserts that this list exhausts \(\widehat{G}\) (here a direct approach is used, without undue attention paid to the non- unitary dual). In the course of his exposition, the author also gives a nice survey of general representation theory, including even definitions of such basic notions as Haar measure, algebraic groups, \(p\)-adic numbers, the unitary dual, Jacquet modules, etc. Therefore this nicely written paper is to be recommended to non-experts as well as experts on the field.
Reviewer: S.Gelbart (Rehovot)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |
Keywords:
irreducible unitary representations; non-archimedean local field; survey of general representation theoryCitations:
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