Character Sums Asociated to Prehomogeneous Vector Spaces

Abstract

Let G be a complex linear algebraic group and ρ:G → GL(V) a finite dimensional rational representation. Assume that G is connected and reductive, and that V has an open G-orbit. Let f in C[V] be a non-zero relative invariant with character φ ∈ Hom (G, C^×), meaning that f ^ ρ (g) =φ (g) f for all g in G. Choose a non-zero relative invariant f^v in C[V^v], with character φ^-1, for the dual representation ρ^v:G → GL(V^v). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sato says that the Fourier transform of |f|^s equals |f^v>|^>-s up to some factors. The purpose of the present paper is to study a finite field analogue of Sato's theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field $\mathbb{F}_q$ is large enough. Now |f|^s is replaced by χ (f), with χ in Hom ($\mathbb{F}$ ^×_q , ℂ^×), and the factors involve Gauss sums, the Bernstein–Sato polynomial b(s) of f, and the parity of the split rank of the isotropy group at v^v>∈ V^v($\mathbb{F}_q$). We also express this parity in terms of the quadratic residue of the discriminant of the Hessian of log f^v (v^v). Moreover we prove a conjecture of N. Kawanaka on the number of integer roots of b(s).