Degenerate principal series for even-orthogonal groups

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 Levi factor $M \cong GL(k,F) \times O(2(n-k),F)$ (resp. $M \cong GL(k,F) \times SO(2(n-k),F)$). We consider a one-dimensional representation of $M$ of the form $\chi \circ det_k \otimes triv_{(n-k)}$, with $\chi$ a one-dimensional representation of $F^{\times}$; this may be extended trivially to get a representation of $P$. We consider representations of the form $\mbox{Ind}_P^G(\chi \circ det_k \otimes triv_{(n-k)}) \otimes 1$. (Our results also work when $G=O(2n,F)$ and the inducing representation is $(\chi \circ det_k \otimes det_{(n-k)}) \otimes 1$, 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 $SO(2n,F)$.