Fourier transform as a triangular matrix

Abstract

Let V V be a finite dimensional vector space over the field with two elements with a given nondegenerate symplectic form. Let [ V ] [V] be the vector space of complex valued functions on V V , and let [ V ] Z [V]_{\mathbf {Z}} be the subgroup of [ V ] [V] consisting of integer valued functions. We show that there exists a Z \mathbf {Z} -basis of [ V ] Z [V]_{\mathbf {Z}} consisting of characteristic functions of certain isotropic subspaces of V V and such that the matrix of the Fourier transform from [ V ] [V] to [ V ] [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.