Degenerate principal series for symplectic and odd-orthogonal groups

Let F be a p-adic field and G = SO2n+1(F) (resp. Sp(2n)(F)). A maximal parabolic subgroup of G has the form P = MU, with Levi factor M congruent to GL(k)(F) x SO2(n-k+1(F) (resp. M congruent to GL(k)(F)) x Sp(2(n-k))(F)). A one-dimensional representation of M has the form chi circle det(k) x triv((n-k)), with chi a one-dimensional representation of F-x; this may be extended trivially to get a representation of P. We consider representations of the form Ind(p)(G) (chi circle det(k) x triv((n-k))) x 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 less than or equal to 4.) Our approach is based on Jacquet module techniques developed by M. Tadic.