Akihiko Gyoja

Character Sums Asociated to Prehomogeneous Vector Spaces

AbstractLet 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 fv in C[Vv], with character φ-1, for the dual representation ρv:G → GL(Vv). Roughly, the fundamental theorem of the theory of prehomogeneous vector spaces due to M. Sato says that the Fourier transform of |f|s equals |fv>|>-s up to some factors. The purpose of the present paper is to study a finite field analogue of Satos theorem and to give a completely explicit description of the Fourier transform assuming that the characteristic of the base field

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

\mathbb{F}_q

Modular Representation Theory over a Ring of Higher Dimension with Applications to Hecke Algebras

is large enough. Now |f|s is replaced by χ (f), with χ in Hom (

Associated variety, Kostant-Sekiguchi correspondence, and locally free

\mathbb{F}

-action on Harish-Chandra modules

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 vv>∈ Vv(