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Geometry of dual spaces of reductive groups (non-Archimedean case). (English) Zbl 0664.22010
Let $$\tilde G$$ (resp. Ĝ) denote the set of equivalence classes of irreducible smooth (resp. unitarizable) representations of a connected reductive group G over a nonarchimedean local field F; this is the so- called dual (resp. unitary dual) space of G. Several years ago, the author (and also D. Miličić) made a careful study of the topology of these dual groups, and their relation to properties of the representation theory of G [Glas. Mat., III. Ser. 18(38), 259-279 (1983; Zbl 0536.22026); also ibid. 8(28), 7-22 (1973; Zbl 0265.46072)]. More recently, the author gave a detailed treatment of these questions for the group $$G=GL(n,F)$$, building on the Bernstein-Zelevinsky parametrization of $$\tilde G,$$ and the author’s own parametrization of $$\hat G$$ [Duke Math. J. 55, 385-422 (1987)]; typical results there included a classification of all “isolated points modulo center” in $$\hat G,$$ description of the composition factors of ends of complementary series representations, and a description of $$\hat G$$ as an abstract topological space. In the present paper, the author reproves (and improves) some of his (and Miličić’s) earlier general results on the topology of $$\tilde G$$ and $$\hat G,$$ this time using the previously unavailable “Bernstein center”. The paper covers much important basic material, and provides a useful contrast between $$\hat G$$ and $$\tilde G.$$
Reviewer: S.Gelbart

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
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