Functorial Zeta Integrals

TL;DR

This paper reinterprets the proof of the functional equation for nonarchimedean Rankin-Selberg local Euler factors using functorial language, enhancing clarity and generality over different fields.

Contribution

It provides a functorial reformulation of the original proof, making the concepts clearer and applicable over arbitrary fields with suitable characteristics.

Findings

Clearer presentation of the functional equation proof

Extension of the proof to arbitrary fields

Use of Bernstein-Zelevinsky functorial language

Abstract

The functional equation for nonarchimedean Rankin-Selberg local Euler factors was proved by Jacquet, Piatetski-Shapiro, and Shalika in 1983. In this expository note we translate the original proof into the purely functorial language of parabolic induction-restriction of Bernstein-Zelevinsky. This new language gives a clearer presentation of the ideas, and works over arbitrary fields with characteristic not equal to the residue characteristic.