Spherical Functions on 2-adic Ramified Hermetian Spaces

TL;DR

This paper explicitly computes spherical functions on 2x2 Hermitian matrices over 2-adic ramified quadratic extensions, completing Hironaka's work by addressing the previously unconsidered ramified case.

Contribution

It extends the explicit computation of spherical functions to the ramified case of 2-adic Hermitian spaces, which was not previously addressed.

Findings

Explicit formulas for spherical functions in the ramified case

Functional equations for these spherical functions

Completion of Hironaka's analysis for all ramification cases

Abstract

Y. Hironaka introduced the spherical functions on the p-adic space of Hermitian matrices. For the space of 2\times2 Hermitian matrices, we complete Hironaka's work by also considering the case of a wildly ramified quadratic extension. We compute the spherical functions explicitly and obtain the functional equations.