On supports of induced representations for symplectic and odd-orthogonal groups

Abstract

Let G be Sp (2 n, F ) (resp. SO (2 n + 1, F )), where F is a p -adic field of characteristic zero. In this paper, we give a correspondence which associates to an irreducible representation π of G an m -tuple of irreducible representations of lower rank symplectic (resp. orthogonal) groups based on the supercuspidal support of π. We show that this correspondence respects the induction and Jacquet module functors (in a sense to be made precise), as well as verifying a number of other useful properties. In essence, this correspondence allows one to isolate the effects of the different families of supercuspidal representations of general linear groups which appear in the support of π.

Jacquet modules of 𝑝-adic general linear groups

“…The first claim is an immediate consequence of Proposition 8.5 of [Zel]. Claims (2) and (3) follow fairly easily; see Corollary 5.6 and section 10 of [Jan2] for details.…”

“…We let m * X π be the sum of everything in m * π of the form τ ⊗ θ with τ, θ irreducible and r min τ containing a term of the form ν ( Jan2]). Observe that (1) and (2) clearly hold.…”

“…This follows from [Zel] or an argument like that in Proposition 5.3.2 of [Jan4] (but this case is simpler).…”

“…Then, as discussed in section 2.4 of [Jan4], for an irreducible representation π supported on {ν α ρ} α∈Z , we have δ 0 (π) inductively equivalent to the Langlands data for π. For this reason, we shall freely move between δ 0 (π) and the Langlands data for π, referring to both as the Langlands classification and freely using both in the notation for the Langlands subrepresentation.…”

“…Proof. The first claim follows from Lemma 2.4.2 of [Jan4]. Multiplicity one is part of the Langlands classification.…”