Second adjointness and cuspidal supports at the categorical level

TL;DR

This paper establishes second adjointness within the categorical local Langlands framework, explores the link between Eisenstein series and cuspidal supports, and proposes a conjecture on supercuspidal L-parameters.

Contribution

It introduces a proof of second adjointness at the categorical level and offers a conjectural characterization of certain irreducible representations.

Findings

Proved second adjointness in categorical local Langlands

Analyzed relations between Eisenstein series and cuspidal supports

Conjectured a characterization of supercuspidal L-parameters

Abstract

We prove the second adjointness in the setting of the categorical local Langlands correspondence. Moreover, we study the relation between Eisenstein series and cuspidal supports and present a conjectural characterization of irreducible smooth representations with supercuspidal $L$-parameters regarding geometric constant terms. The main technical ingredient is an induction principle for geometric Eisenstein series which allows us to reduce to the situations already treated in the literature.