The Doubling Method in Algebraic Families

Abstract

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 appropriate normalizing factor that 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.