Geometric side of a local relative trace formula

Abstract: Following a scheme suggested by B. Feigon, we investigate a local relative trace formula in the situation of a reductive $p$ -adic group $G$ relative to a symmetric subgroup $H= \underline{H}(F)$ where $\underline{H}$ is split over the local field $F$ of characteristic zero and $G = \underline{G} (F)$ is the restriction of scalars of $\underline{H} _{I E}$ relative to a quadratic unramified extension $E$ of $F$. We adapt techniques of the proof of the local trace formula by J. Arthur in order to get a geometric expansion of the integral over $H \times H$ of a truncated kernel associated to the regular representation of $G$.