The Whittaker-Shintani functions for symplectic groups

TL;DR

This paper derives a formula for Whittaker-Shintani functions on p-adic symplectic groups, generalizing known functions, and uses it to prove a conjecture related to L-functions for symplectic and general linear groups.

Contribution

It provides a new explicit formula for Whittaker-Shintani functions on p-adic symplectic groups and applies it to prove a conjecture on unramified L-function calculations.

Findings

Derived a formula for Whittaker-Shintani functions on p-adic symplectic groups

Provided an alternative proof of Shintani's conjecture on L-functions

Extended the understanding of special functions in representation theory

Abstract

In this note, we give a formula for the Whittaker-Shintani functions for the p-adic symplectic groups, which is a generalization of the Zonal spherical functions and Whittaker functions. We then use the formula to give an alternative proof of a conjecture given by T.Shintani on the unramified calculation of L-function for $Sp_{2n}\times GL_{1}$.