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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:
22E50 Representations of Lie and linear algebraic groups over local fields
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