The Doubling Method in Algebraic Families

28.06.2022 13:15 - 14:45

Johannes Girsch (Imperial College)

Local constants are an important concept in the complex representation theory of reductive $p$-adic groups, for example they are pivotal in the formulation of the Local Langlands correspondence. In recent years there has been progress in defining such constants for modular representations or in even more general settings. For example, Moss was able to define $\gamma$-factors for representations of $\GL_n(\mathbb Q_p)$ with coefficients in general noetherian rings and subsequently together with Helm was able to prove a converse theorem, which was crucial for the proof of the Local Langlands correspondence in families for $\GL_n$. The aim of this talk is to show how one can extend the Doubling Method of Piateski-Shapiro and Rallis to families of representations of classical groups. In this setting we will introduce and prove a rationality result for the Doubling Zeta integrals. Subsequently we will show that these zeta integrals satisfy a functional equation from which one obtains $\gamma$-factors.

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