A note on local periods for supercuspidal representations

Let $G$ be a $p$-adic reductive group and $H$ a unimodular spherical subgroup of $G$. Let $\pi$ be a unitary supercuspidal representation of $G$. In this note, under a mild assumption, we show that local periods in $Ho{m}_H(\pi ,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $\pi$ over $H$.