Spherical character of a supercuspidal representation as weighted orbital integral

Patrick Delorme (1) , Pascale Harinck (2)

1 I2M - Institut de Mathématiques de Marseille
2 CMLS - Centre de Mathématiques Laurent Schwartz

Let $\rm E/\rm F$ 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 $\rm F$. We assume that the split connected component of the center of $\underline{H}$ is trivial. Let $(\tau,V)$ be a $\underline{H}(\rm F)$-distinguished supercuspidal representation of $\underline{H}(\rm E)$. 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 $\underline{H}(\rm F)$-invariant linear forms on $V$ in terms of weighted orbital integrals of matrix coefficients of $\tau$.

Mots clés: p-adic reductive groups; symmetric spaces; truncated kernel; spher- ical character; weighted orbital integral

Domaines: Théorie des représentations [math.RT]

Patrick Delorme, Pascale Harinck. Spherical character of a supercuspidal representation as weighted orbital integral. 2016. ⟨hal-01493259⟩