The Twisted Doubling Method in Algebraic Families

TL;DR

This paper introduces twisted doubling zeta integrals in algebraic families, proves their rationality and functional equations, and defines associated gamma factors for classical and general linear group representations.

Contribution

It extends the twisted doubling method to algebraic families, establishing foundational properties and defining gamma factors in this broader context.

Findings

Proved rationality of twisted doubling zeta integrals.

Established functional equations for these integrals.

Defined gamma factors for families of representations.

Abstract

We define the twisted doubling zeta integrals of Cai-Friedberg-Ginzburg-Kaplan in the setting of algebraic families. We then prove a rationality result and a functional equation for these zeta integrals. This allows us to define an unnormalized $\gamma$-factor associated to certain families of representation of a classical group times a general linear group.