Abstract
We extend the results of Emerton on the ordinary part functor to the category of the smooth representations over a general commutative ring R, of a general reductive p-adic group G (rational points of a reductive connected group over a local non-archimedean field F of residual characteristic p). In Emerton’s work, the characteristic of F is 0, R is a complete artinian local \(\mathbb{Z}_{p}\) -algebra having a finite residual field, and the representations are admissible. We show:
The smooth parabolic induction functor admits a right adjoint. The center-locally finite part of the smooth right adjoint is equal to the admissible right adjoint of the admissible parabolic induction functor when R is noetherian. The smooth and admissible parabolic induction functors are fully faithful when p is nilpotent in R.
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Notes
- 1.
This work is now written in N. Abe, Noriyuki, G. Henniart, F. Herzig, M.-F. Vignéras - A classification of admissible irreducible modulo p representations of reductive p-adic groups. To appear in Journal of the A.M.S. 2016.
- 2.
We know now that R G P respects the admissibility (N. Abe, Noriyuki, G. Henniart, F. Herzig, M.-F. Vignéras - Mod p representations of reductive p-adic groups: functorial properties. In progress, 2016.)
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Vignéras, MF. (2016). The Right Adjoint of the Parabolic Induction. In: Ballmann, W., Blohmann, C., Faltings, G., Teichner, P., Zagier, D. (eds) Arbeitstagung Bonn 2013. Progress in Mathematics, vol 319. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43648-7_15
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DOI: https://doi.org/10.1007/978-3-319-43648-7_15
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