Spherical functions on $p$-adic homogeneous spaces

TL;DR

This paper provides a general formula for spherical functions on p-adic homogeneous spaces, introduces a unified method for their functional equations, and connects these to p-adic local zeta functions of small prehomogeneous vector spaces.

Contribution

It introduces a comprehensive expression for spherical functions on p-adic spaces and links their functional equations to those of p-adic zeta functions, unifying existing approaches.

Findings

Derived a general expression for spherical functions.

Established a method to obtain functional equations.

Connected spherical functions to p-adic local zeta functions.

Abstract

We give a general expression of spherical functions on $p$-adic homogeneous spaces of $G$, based on data of $G$ and functional equations of spherical functions. Then, we show a unified method to obtain functional equations of spherical functions and relate them those of $p$-adic local zeta functions of small prehomogeneous vector spaces.