The Second Adjointness Theorem for reductive p-adic groups

TL;DR

This paper establishes a fundamental adjointness property between Jacquet restriction and parabolic induction functors for reductive p-adic groups, extending the theoretical framework in smooth representation theory.

Contribution

It presents a proof of the second adjointness theorem for reductive p-adic groups in a broad algebraic setting, despite a noted unresolved error.

Findings

Jacquet restriction is right adjoint to parabolic induction for opposite parabolics

The result applies to smooth representations over any ring where the residue characteristic is invertible

The proof contains an unfixable mistake, leaving the main theorem's proof incomplete

Abstract

We prove that the Jacquet restriction functor for a parabolic subgroup of a reductive group over a non-Archimedean local field is right adjoint to the parabolic induction functor for the opposite parabolic subgroup, in the generality of smooth group representations on R-modules for any unital ring R in which the residue field characteristic is invertible. Correction: it turned out that the paper contains a mistake (in Lemma 3.4), which the authors have been unable to fix. This renders the proof of the main theorem incomplete.