The Doubling Method in Algebraic Families

TL;DR

This paper introduces the doubling zeta integral for classical group representations, proves its rationality and functional equation, and constructs gamma factors that generalize existing definitions within algebraic families.

Contribution

It develops a new framework for gamma factors in algebraic families of representations, extending classical results to a broader, family-based context.

Findings

Proves rationality of the doubling zeta integral.

Establishes a functional equation for these integrals.

Constructs gamma factors compatible with classical definitions.

Abstract

We define the doubling zeta integral for smooth families of representations of classical groups. Following this we prove a rationality result for these zeta integrals and show that they satisfy a functional equation. Moreover, we show that there exists an apropriate normalizing factor which allows us to construct $\gamma$-factors for smooth families out of the functional equation. We prove that under certain hypothesis, specializing this $\gamma$-factor at a point of the family yields the $\gamma$-factor defined by Piateski-Shapiro and Rallis.