Universal hierarchical structure of reducibility of Harish-Chandra parabolic induction

TL;DR

This paper uncovers a universal hierarchical structure governing the reducibility of parabolic induction in reductive groups, providing new proofs and confirming conjectures related to irreducibility and finiteness of special exponents.

Contribution

It introduces a universal hierarchical framework for reducibility of parabolic induction, offering new proofs and confirming conjectures in the representation theory of reductive groups.

Findings

Established a universal hierarchical structure of reducibility

Provided a new proof of generic irreducibility of parabolic induction

Proved Clozel's finiteness conjecture under certain conditions

Abstract

Given a supercuspidal representation $\sigma$ of a parabolic subgroup $P$ of reductive group $G$, we discover a universal hierarchical structure of reducibility of the parabolic induction $Ind^G_P(\sigma)$, i.e. always irreducible from some Levi-level up. As its applications, we provide a new simple proof of the generic irreducibility property of parabolic induction, and prove Clozel's finiteness conjecture of special exponents under some conditions. Indeed, those conditions are predicted by two conjectures of Shahidi which in some sense are proved for classical groups by Arthur in his monumental book--The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups. At last, naturally, such type simple beautiful structure theorem should be conjectured to hold in general, i.e. if the "reducibility conditions" of a general parabolic induction lies in some Levi subgroup, then it is always irreducible from this Levi up.