# Second adjointness for tempered admissible representations of a real group

## TL;DR

This paper extends the concept of second adjointness to tempered admissible representations of real reductive groups, generalizing prior results from SL_2 to broader groups and exploring related pairings and functors.

## Contribution

It generalizes second adjointness from SL_2 to all real reductive groups for tempered admissible representations and discusses related pairings and functors.

## Key findings

- Generalization of second adjointness to broader groups
- Analysis of Casselman's canonical pairing
- Discussion of Bernstein morphisms and functors

## Abstract

We study second adjointness in the context of tempered admissible representations of a real reductive group. Compared to a recent result of Crisp and Higson, this generalizes from $SL_2$ to a general group, but specializes to only considering admissible representations. We also discuss Casselman's canonical pairing in this context, and the relation to Bernstein morphisms. Additionally, we take the opportunity to discuss some relevant functors and some of their relations.