Fourier transform as a triangular matrix

Let V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [V] be the vector space of complex valued functions on V and let [V]_Z be the subgroup of [V] consisting of integer valued functions. We show that there exists a Z-basis of [V]_Z consisting of characteristic functions of certain explicit isotropic subspaces of V such that the matrix of the Fourier transform from [V] to [V] with respect to this basis is triangular. We show that this is a special case of a result which holds for any two-sided cell in a Weyl group.