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The Twisted Doubling Method in Algebraic Families

Johannes Girsch

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 $γ$-factor associated to certain families of representation of a classical group times a general linear group.

The Twisted Doubling Method in Algebraic Families

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 -factor associated to certain families of representation of a classical group times a general linear group.

Paper Structure

This paper contains 22 sections, 37 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Background in Representation Theory
  4. Hecke Algebras
  5. Finiteness Results
  6. $\ell$-Sheaves and Cosmooth Modules
  7. $\ell$-Sheaves on $D^r$
  8. Degenerate Whittaker Models
  9. Invariance under stabiliser
  10. Classical Groups
  11. The reduced adjugate matrix
  12. Setup of the Twisted Doubling Method
  13. Parabolic Subgroups
  14. The Character $\psi^\bullet$
  15. The Twisted Doubling Zeta Integrals
  16. ...and 7 more sections

Key Result

Theorem A

If the contragredient $\widetilde{\pi}$ of $\pi$ is admissible and finitely generated as an $A[G]$-module there is a polynomial $\mathcal{P}\in S_{A\otimes_R B}$ such that for any $v\in\pi,\lambda\in\widetilde{\pi}$ and $f\in I(X,\theta)$.

Theorems & Definitions (77)

  • Theorem A: Theorem \ref{['rattheo']}
  • Theorem B: Theorem \ref{['funcequ']}
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 67 more