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.

In this paper, we prove the Langlands quotient theorem in the context of finite central extensions of connected, reductive p-adic groups.

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.

Let G be a classical p-adic group and let π be a smooth irreducible representation of G. In this paper, we consider the problem of calculating the dual (in the sense of Aubert and Schneider-Stuhler)π. More precisely, if π is specified by its Langlands data, the problem is to determine the Langlands data forπ. This problem reduces (based on supercuspidal support) to two main cases: half-integral reducibility and integral reducibility; the latter is addressed here.

Let F be a p-adic field of characteristic 0 and G = O(2n, F ) (resp. SO(2n, F )). A maximal parabolic subgroup of G has the form P = MU, with, 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(Our results also work when G = O(2n, F ) and the inducing representation is (χ•det k ⊗det (n−k) )⊗1, using det (n−k) to denote the nontrivial character of O(2(n − k), F ).) More generally, we allow Zelevinsky segment representations for the inducing representations.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. For O(2n, F ), we use Jacquet module methods to obtain our results; the results for SO(2n, F ) are obtained via an analysis of restrictions to SO(2n, F ).