Spherical character of a supercuspidal representation as weighted orbital integral

Abstract

Let $E/\mathrm{F}$ be an unramified quadratic extension of local non archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic reductive group, defined and split over $\mathrm{F}$. We assume that the split connected component of the center of $\underline{H}$ is trivial. Let $(\ddot{I}„,V)$ be a $\underline{H}(\mathrm{F})$-distinguished supercuspidal representation of $\underline{H}(\mathrm{E})$. Using the recent results of C. Zhang, and the geometric side of a local relative trace formula obtained by P. Delorme, P. Harinck and S. Souaifi, we describe spherical characters associated to $\underline{H}(\mathrm{F})$-invariant linear forms on $V$ in terms of weighted orbital integrals of matrix coefficients of $\ddot{I}„$.

7 pages