Local zeta functions for a class of p-adic symmetric spaces (II)

TL;DR

This paper investigates zeta functions linked to minimal spherical principal series representations of certain p-adic symmetric spaces, establishing explicit functional equations and defining associated L-functions and epsilon-factors for a specific subclass.

Contribution

It provides explicit functional equations for zeta functions of p-adic symmetric spaces and introduces L-functions and epsilon-factors for a subclass, extending previous studies.

Findings

Zeta functions satisfy explicit functional equations.

Defined L-functions and epsilon-factors for a subclass.

Extended understanding of p-adic symmetric space representations.

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.