Spherical functions on $U(n,n)/(U(n) \times U(n))$ and hermitian Siegel series

TL;DR

This paper investigates spherical functions on a specific symmetric space over p-adic fields, deriving functional equations, explicit formulas, and applying these results to establish a functional equation for p-adic hermitian Siegel series.

Contribution

It provides explicit formulas and functional equations for spherical functions on $U(n,n)/(U(n) imes U(n))$ and applies these to hermitian Siegel series.

Findings

Derived functional equations for spherical functions

Identified poles and zeros of the functions

Established a functional equation for p-adic hermitian Siegel series

Abstract

We study spherical functions on the space isomorphic to $U(2n)/(U(n)\times U(n))$ over a $p$-adic field; those functional equations with respect to the action of the Weyl group, the location of possible poles and zeros, explicit formulas, and spherical Fourier transforms. Then, as an application, we give a functional equation of $p$-adic local hermitian Siegel series.