An adjunction formula for the Emerton-Jacquet functor

TL;DR

This paper establishes an adjunction formula for the Emerton-Jacquet functor, linking it to locally analytic inductions under non-critical conditions, and explores its relation to socles of principal series.

Contribution

It introduces a new adjunction formula for the Emerton-Jacquet functor and analyzes its connection to locally analytic inductions and socles in the non-critical case.

Findings

Derived an adjunction formula relating Emerton-Jacquet functor to locally analytic inductions.

Analyzed the relationship between the functor and socles of principal series.

Provided conditions under which the formula applies, called non-critical hypothesis.

Abstract

The Emerton-Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton-Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.