Degenerate principal series for symplectic and odd-orthogonal groups

Abstract

Let F be a p-adic field and G = SO 2n+1 (F) (resp. Sp 2n (F)). A maximal parabolic subgroup of G has the form P = MU, with Levi factor M ∼ = GL k (F) × SO 2(n−k)+1 (F) (resp. M ∼ = GL k (F) × Sp 2(n−k) (F)). A one-dimensional representation of M has the form χ • det k ⊗ triv (n−k) , with χ a one-dimensional representation of F × ; this may be extended trivially to get a representation of P. We consider representations of the form Ind G P (χ • det k ⊗ triv (n−k)) ⊗ 1. (More generally, we allow Zelevinsky segment representations for the inducing representation.) In this paper, we study the reducibility of such representations. We determine the reducibility points, give Langlands data and Jacquet modules for each of the irreducible composition factors, and describe how they are arranged into composition series. (Note: it turns out that the composition series has length ≤ 4.) Our approach is based on Jacquet module techniques developed by M. Tadić.

A theory of Miyawaki liftings: the Hilbert–Siegel case

“…To show Theorem 3.1 (2), we need to prove the irreducibility of the induced representation (µ • det n−r ) ⋊ π, where π is an irreducible almost tempered representation of Sp r (F ), and µ is a unitary character of F × . When π is supercuspidal and δ = 0, the irreducibility of (µ • det n−r ) ⋊ π was proven by Tadić [45, Theorem 9.1], and its Langlands data was given by Jantzen [30]. We imitate their proofs.…”

Degenerate principal series and Langlands classification

“…Remark 3.2. One can derive analogous results for symplectic and split even orthogonal groups of even rank from [10,2,12] (cf. Theorem 8.2 of [19]).…”

On Square-Integrable Representations of Classical p-adic Groups

“…generally seem to be enough to determine how induced representations supported on S (ρ, β); σ decompose, especially when Jacquet module methods are employed (cf. [Tad3], [Tad4], [Tad5], [Jan1], [Jan2]).…”