Generalization of a theorem of Waldspurger to nice representations

A theorem of Waldspurger states that the Fourier transform of a stable distribution on the Lie algebra of a simply-connected semisimple group $G$ over a p-adic field, is again stable. We generalize this theorem to representations whose generic stabilizer subgroup is connected and reductive (assuming that $G$ is simple). In this more general situation the Fourier transform of a stable distribution is stable up to a sign that we describe explicitly. The proof is based on the $p$-adic stationary phase principle and on the global techniques introduced by Kottwitz for stabilization of the trace formula. As an application of our main theorem, we find the explicit diagonalization of the gamma-matrix for the prehomogeneous space of symmetric $n\times n$ matrices over a p-adic field (for odd $n$).