Weight filtrations for induced representations of real reductive Lie groups (Q1122670)

Let G be a connected semisimple Lie group with finite center and \(P=MAN\) a parabolic subgroup of G. If \(\pi\) is an admissible representation of M and \(e^{\nu}\) a character of A, one can define the (normalized) induced representation \(I_ P(\pi \otimes e^{\nu})\) of G. If the infinitesimal character of this representation is regular integral and \(\pi\) is irreducible, the authors construct a filtration of the induced representation such that the associated graded module is semisimple. The filtration is defined in geometric terms, using the localization functor of \textit{A. A. Beilinson} and \textit{J. N. Bernstein} [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)], the Riemann-Hilbert correspondence between \({\mathcal D}\)-modules and perverse sheaves, and the transfer to positive characteristics [\textit{A. A. Beilinson}, \textit{J. N. Bernstein}, \textit{P. Deligne}, ``Faisceaux pervers'' (Astérisque 100, 1982; Zbl 0536.14011)]. In the case of a cuspidal parabolic subgroup P, a discrete series representation \(\pi\) and a ``negative'' \(\nu\), this filtration agrees with the Jantzen filtration [\textit{A. A. Beilinson}, Proc. Int. Congr. Math., Warszawa 1983, 699-710 (1984; Zbl 0571.20032)].