FOURIER TRANSFORM AS A TRIANGULAR MATRIX

Abstract

©2020 American Mathematical Society abstract. 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 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.