Local zeta functions for a class of p-adic symmetric spaces (II)
Pascale Harinck, Hubert Rubenthaler
TL;DR
The paper develops a comprehensive framework for local zeta theory on a class of p-adic symmetric spaces by extending the structural analysis of regular graded Lie algebras (Part I) to the study of zeta integrals attached to the minimal spherical principal series (Part II). It builds on a detailed descent along strongly orthogonal roots to describe $G$-orbits in the open $V^+$, then leverages prehomogeneous vector space methods to establish rationality and an explicit functional equation for zeta integrals, including an explicit gamma factor. In special cases it constructs L-functions and epsilon-factors, generalizing Godement–Jacquet type results to a broad family of p-adic symmetric spaces. The work integrates root-system combinatorics, invariant theory via the fundamental invariant $\Delta_0$, and explicit orbit classifications to enable explicit zeta-functional equations with potential applications to automorphic and representation-theoretic problems in the p-adic setting.
Abstract
In this paper we study the zeta functions associated to the minimal spherical principal series of representations for a class of reductive p-adic symmetric spaces, which are realized as open orbits of some prehomogeneous spaces. These symmetric spaces have been studied in the paper arXiv: 2003.05764. We prove that the zeta functions satisfy a functional equation which is given explicitly (see Theorem 4.3.9 and Theorem 4.4.5). Moreover, for a subclass of these spaces, we define L-functions and epsilon-factors associated to the representations.