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 $[V]_{\mathbf {Z}}$ be the subgroup of $[V]$ consisting of integer valued functions. We show that there exists a $\mathbf {Z}$-basis of $[V]_{\mathbf {Z}}$ consisting of characteristic functions of certain isotropic subspaces of $V$ and 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.