On the convergence of zeta functions of prehomogeneous vector spaces

TL;DR

This paper establishes a broad convergence theorem for zeta functions associated with prehomogeneous vector spaces, extending prior results and analyzing boundary contributions, especially in the context of nilpotent orbits.

Contribution

It provides a general convergence criterion for zeta functions of prehomogeneous vector spaces and explores boundary terms and residues in specific cases.

Findings

Proves a general convergence result for zeta functions.

Identifies boundary terms related to certain subspaces.

Determines residues at the rightmost pole in some cases.

Abstract

We prove a general convergence result for zeta functions of prehomogeneous vector spaces extending results of H. Saito, F. Sato and Yukie. Our analysis points to certain subspaces which yield boundary terms. We study it further in the setup arising from nilpotent orbits. In certain cases we determine the residue at the rightmost pole of the zeta function.