Character sums and intersection cohomology complexes associated to the space of square matrices
The author studies Fourier transforms of relatively invariant functions on the square matrix spaces and Hermitian matrix spaces over the finite field \(\mathbb{F}_q\), which are typical prehomogeneous vector spaces over a finite field. In the general setting, the formula of Fourier transform of a relatively invariant function on a prehomogeneous vector space was already obtained by the author provided that the characteristic of \(\mathbb{F}_q\) is sufficiently large and that we consider only the open orbit of the prehomogeneous vector space. One purpose of this paper is to exclude the constraint in these special examples and give an explicit formula of the Fourier transform. Though in this paper we only consider two examples, this is the first step to the study of prehomogeneous vector spaces using intersection cohomology theory.
Fourier transforms of relatively invariant functions
scientific article; zbMATH DE number 1136777