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| ≠ 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 μ-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.
J.-F. Dat. "ν-tempered representations of p-adic groups, I: l-adic case." Duke Math. J. 126 (3) 397 - 469, 15 February 2005. https://doi.org/10.1215/S0012-7094-04-12631-4