Dat, J.-F. \(\nu\)-tempered representations of \(p\)-adic groups. I: \(l\)-adic case. (English) Zbl 1063.22017 Duke Math. J. 126, No. 3, 397-469 (2005). Summary: The so-called tempered complex smooth representations of \(p\)-adic groups have been much studied and used, in connection with automorphic forms. Nevertheless, the smooth representations that are realized geometrically often have \(l\)-adic coefficients, so that Archimedean estimates of their matrix coefficients hardly make sense. We investigate here a notion of tempered representation with coefficients in any normed field of characteristic not equal to \(p\). The theory turns out to be different according to the norm being Archimedean, non-Archimedean with \(|p|\neq 1\), or non-Archimedean with \(|p|= 1\). In this paper we concentrate on the last case. The main applications concern modular representation theory (i.e., on a positive characteristic field) and, in particular, the study of reducibility properties of the parabolic induction functors; one of the main results is the generic irreducibility for induced families. Thanks to a suitable theory of rational intertwining operators, this allows us to define Harish-Chandra’s \(\mu\)-functions and show in some special cases how they track down the cuspidal constituents of parabolically induced representations. Besides, we discuss the admissibility of parabolic restriction functors and derive some lifting properties for supercuspidal modular representations. Cited in 1 ReviewCited in 14 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields 20G05 Representation theory for linear algebraic groups × Cite Format Result Cite Review PDF Full Text: DOI References: [1] I. N. [J.] Bernšteĭn, All reductive \(\mathfrakp\)-adic groups are of type I (in Russian), Funkcional Anal. i Priložen. 8 , no. 2 (1974), 3-6; English translation in Funct. Anal. Appl. 8 (1974), 91-93. [2] J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive \(p\)-adic groups , J. Analyse Math. 47 (1986), 180-192. · Zbl 0634.22011 · doi:10.1007/BF02792538 [3] J. N. Bernstein, P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des groupes réductifs sur un Corps local , Travaux en Cours, Hermann, Paris, 1984. · Zbl 0544.00007 [4] I. N. [J.] Bernstein and A. V. 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