Universal hierarchical structure of reducibility of Harish-Chandra parabolic induction

2019/03/23 by Caihua Luo, Luo, Caihua
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1903.09774

openalex publication_date 2019/03/23 · openalex created_date 2019/10/18 · openalex updated_date 2026/07/28

Abstract

Given a supercuspidal representation σ of a parabolic subgroup P of reductive group G, we discover a universal hierarchical structure of reducibility of the parabolic induction Ind^G_P(σ), 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.

Citations

• Unitary dual of quasi-split PGSO₈^E
• Muller type irreducibility criterion for generalized principal series

Related

• Generic irreducibility of parabolic induction for real reductive groups
• Orbites unipotentes et poles d'ordre maximal de la fonction μ de Harish-Chandra
• On reducibility of induced representations of odd unitary groups: the depth zero case
• On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions
• Comparison of compact induction with parabolic induction