Relative Non-cuspidality of Representations Induced from Split Parabolic Subgroups

Abstract

Let $G/H$ be a reductive symmetric space over a $p$-adic field, defined by an involution $\sigma$ on $G$. We study the induced representations ${\rm Ind}_P^G(\rho)$ for any proper $\sigma$-split parabolic subgroup $P=MU$ and any $M\cap H$-distinguished representation $\rho$ of $M$. Let ${\mathcal X}$ denote the complex torus of unramified characters of $M$ which are trivial on $M\cap H$ and $\rho_{\chi}$ the twist of $\rho$ by $\chi\in {\mathcal X}$. We show that ${\rm Ind}_P^G(\rho_{\chi})$ is not $H$-relatively cuspidal {\it for generic} $\chi$, that is, for $\chi$ in a Zariski-open subset of ${\mathcal X}$.