Reducibility of representations induced from the Zelevinsky segment and discrete series

Abstract

Let $$G_n$$ denote either the group $$SO(2n+1, F)$$ or Sp(2n, F) over a non-archimedean local field. We determine the reducibility criteria for a parabolically induced representation of the form $$\langle \Delta \rangle \rtimes \sigma $$ , where $$\langle \Delta \rangle $$ stands for a Zelevinsky segment representation of the general linear group and $$\sigma $$ stands for a discrete series representation of $$G_n$$ , in terms of the Mœglin-Tadic classification.