Zelevinsky segment and strongly positive representations

Source abstract

Abstract Let G n $G_n$ upper G Subscript n denote either the group S O ( 2 n + 1 , F ) $SO(2n+1, F)$ upper S upper O left parenthesis 2 n plus 1 comma upper F right parenthesis or S p ( 2 n , F ) $Sp(2n, F)$ upper S p left parenthesis 2 n comma upper F right parenthesis over a non-Archimedean local field of characteristic zero. We determine all composition factors of a parabolically induced representation of the form ⟨ [ ν a ρ , ν b ρ ] ⟩ ⋊ σ $\langle [ \nu ^{a} \rho , \nu ^{b} \rho ] \rangle \rtimes \sigma$ left angle bracket left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket right angle bracket right normal factor semidirect product sigma , where ⟨ [ ν a ρ , ν b ρ ] ⟩ $\langle [ \nu ^{a} \rho , \nu ^{b} \rho ] \rangle$ left angle bracket left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket right angle bracket denotes the Zelevinsky segment representation of the general linear group attached to the segment [ ν a ρ , ν b ρ ] $[ \nu ^{a} \rho , \nu ^{b} \rho ]$ left bracket nu Superscript a Baseline rho comma nu Superscript b Baseline rho right bracket , and σ $\sigma$ sigma denotes a strongly positive discrete series of G n $G_n$ upper G Subscript n .