Aubert duals of discrete series: the first inductive step

Abstract

Let Gn denote either symplectic or odd special orthogonal group of rank n over a non-archimedean local field F . We provide an explicit description of the Aubert duals of irreducible representations of Gn which occur in the first inductive step in the realization of discrete series representations starting from the strongly positive ones. Our results might serve as a pattern for determination of Aubert duals of general discrete series of Gn and should produce an interesting part of the unitary dual of this group. Furthermore, we obtain an explicit form of some representations which are known to be unitarizable.

Degenerate principal series for classical and odd GSpin groups in the general case

“…If α = 1 2 , by [22,Lemma 4.10] we have L(δ([ν −α ρ, ν α−1 ρ]); δ(ρ, α; σ)) ∼ = τ (ρ, σ). If α > 1 2 , in the same way as before we get…”

“…Note that both description of subquotients of δ([ν a ρ 0 , ν b ρ 0 ]) δ(ρ, x; σ) and their Aubert duals depend on the reduciblity point β of ρ 0 and σ [22,26]. The description of the Aubert duals happens to be slightly different in the case β = 0.…”

“…The description of the Aubert duals happens to be slightly different in the case β = 0. Accordingly we also consider two cases: Section 3 is the case β = 0 (Section 5 of [22]) and Sections 4, 5, 6 is the case β > 0 (Section 4 of [22]).…”

“…The Aubert dual of the degenerate principal series is a special type of the generalized principal series, and the composition factors of such representations have been determined in [26] and [19,Proposition 3.2]. To determine the Aubert duals of composition factors in question, we use a further adjustment of the methods initiated in [20][21][22]. Eventually, it turns out that needed Aubert duals of tempered representations mostly follow directly from [20,22].…”

“…To determine the Aubert duals of composition factors in question, we use a further adjustment of the methods initiated in [20][21][22]. Eventually, it turns out that needed Aubert duals of tempered representations mostly follow directly from [20,22]. On the other hand, to determine the Aubert duals of the involved non-tempered representations we use an inductive approach based on the detailed investigation of embeddings and Jacquet modules of such representations, using a case-by-case consideration.…”