An explicit inversion formula for the $p$-adic Whittaker transform on $\text{GL}_n(\mathbb{Q}_p)$

TL;DR

This paper derives an explicit inversion and Plancherel formula for the $p$-adic Whittaker transform on $ ext{GL}_n(Q_p)$, enabling new integral representations for local factors of symmetric power $L$-functions.

Contribution

It provides the first explicit inversion and Plancherel formulas for the $p$-adic Whittaker transform on $ ext{GL}_n(Q_p)$, extending classical results.

Findings

Explicit inversion formula for the $p$-adic Whittaker transform

Plancherel formula for the $p$-adic Whittaker transform

Integral representations for local factors of symmetric power $L$-functions

Abstract

Whittaker functions on non-archimedean fields were first introduced in the work of Jacquet, and they were characterized explicitly by Shintani. We obtain an explicit inversion formula and a Plancherel formula for the $p$-adic Whittaker transform on $\text{GL}_n(\mathbb{Q}_p)$. As an application, integral representations are obtained for the local factors of certain symmetric power $L$-functions.