On the functional equation of the Siegel series

TL;DR

This paper proves the functional equation of the Siegel series over non-archimedean local fields using a representation theoretic approach, extending known results from quadratic forms over p-adic fields.

Contribution

It establishes the functional equation of the Siegel series in a broader setting using Kudla and Sweet's representation theoretic methods.

Findings

Proved the functional equation of the Siegel series over non-archimedean local fields.

Extended the known results from quadratic forms over $\\mathbb{Q}_p$ to more general non-archimedean fields.

Utilized representation theoretic techniques to achieve the proof.

Abstract

It is well-known that the Fourier coefficients of Siegel-Eisenstein series can be expressed in terms of the Siegel series. The functional equation of the Siegel series of a quadratic form over $\mathbb{Q}_p$ was first proved by Katsurada. In this paper, we prove the functional equation of the Siegel series over a non-archimedean local field by using the representation theoretic argument by Kudla and Sweet.