The support of closed orbit relative matrix coefficients

Let $F$ be a nonarchimedean local field with odd residual characteristic and let $G$ be the $F$-points of a connected reductive group defined over $F$. Let $\theta$ be an $F$-involution of $G$. Let $H$ be the subgroup of $\theta$-fixed points in $G$. Let $\chi$ be a quasi-character of $H$. A smooth complex representation $(\pi,V)$ of $G$ is $(H,\chi)$-distinguished if there exists a nonzero element $\lambda$ in $\operatorname{Hom}_H(\pi,\chi)$. We generalize a construction of descended invariant linear forms on Jacquet modules first carried out independently by Kato and Takano (2008), and Lagier (2008) to the setting of $(H,\chi)$-distinction. We follow the methods of Kato and Takano, providing a new proof of similar results of Delorme (2010). Moreover, we give an $(H,\chi)$-analogue of Kato and Takano's relative version of the Jacquet Subrepresentation Theorem. In the case that $\chi$ is unramified, $\pi$ is parabolically induced from a $\theta$-stable parabolic subgroup of $G$, and $\lambda$ arises via the closed orbit in $Q\backslash G / H$, we study the (non)vanishing of the descended forms via the support of $\lambda$-relative matrix coefficients.