Symmetry in the functional equation of a local zeta distribution

Abstract

We examine the structure of the coefficient matrix in the functional equation of the zeta distribution of a self-adjoint prehomogeneous vector space over a non-Archimedean local field. Under a restrictive assumption on the generic stabilizers, we show that this matrix is block upper-triangular with almost symmetric blocks; this generalizes a result of Datskovsky and Wright for the space of binary cubic forms.