Character sums and intersection cohomology complexes associated to the space of square matrices

Abstract

Let χ be a multiplicative character of a finite field F q . We study by an algebro-geometric method the finite Fourier transforms of χ(det x), where x runs over the square matrices over F q (untwisted case), or the Hermitian matrices (twisted case). We take up these examples as a first step to develop Intersection Cohomology Method in the theory of prehomogeneous vector spaces.