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 $π$ be a unitary supercuspidal representation of $G$. In this note, under a mild assumption, we show that local periods in $Hom_H(π,\mathbb{C})$ can be constructed by integrating the matrix coefficients of $π$ over $H$.

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