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Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$

Sanjeev Kumar Pandey, C G Venketasubramanian

Abstract

Let $F$ be a non-archimedean local field. For the symplectic group $Sp_{4}(F),$ let $P$ and $Q$ denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions $P=MN$ and $Q=LU.$ For a non-trivial character $ψ$ of the unipotent radical $N$ of $P,$ let $M_ψ$ denote the stabilizer of the character $ψ$ in $M$ under the conjugation action of $M$ on characters of $N.$ For an irreducible representation of the Levi subgroups $M$ or $L,$ let $π$ denote the respective representation of $Sp_{4}(F)$ parabolically induced either from $P$ or from $Q.$ Let $ψ$ be a character of the group $N$ given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module $r_{N,ψ}(π)$ as an $M_ψ$-module. We also deduce the analogous results in the case where $F$ is a finite field of order $q.$

Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$

Abstract

Let be a non-archimedean local field. For the symplectic group let and denote respectively its Siegel and Klingen parabolic subgroups with respective Levi decompositions and For a non-trivial character of the unipotent radical of let denote the stabilizer of the character in under the conjugation action of on characters of For an irreducible representation of the Levi subgroups or let denote the respective representation of parabolically induced either from or from Let be a character of the group given by a rank one quadratic form. In this article, we determine the structure of the twisted Jacquet module as an -module. We also deduce the analogous results in the case where is a finite field of order
Paper Structure (32 sections, 26 theorems, 65 equations)

This paper contains 32 sections, 26 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Generalities on $\ell$-groups
  4. The functors $i_{_{\mathcal{U},1\!\!1}}$ and $r_{\mathcal{U},\theta}$
  5. Symplectic group and its maximal parabolic subgroups
  6. Structure of $P_{k}$
  7. Structure of $N_{k}$
  8. The Siegel parabolic subgroup
  9. The Klingen parabolic subgroup
  10. Preliminary Results
  11. Some general results
  12. Stabilizers of the characters of the unipotent radical $N$ of $P$
  13. Geometric Lemma
  14. Set up for the Geometric Lemma
  15. Orbits for the $S_{\psi}$-action
  16. ...and 17 more sections

Key Result

Theorem 1.1

Fix a non-trivial character $\psi_{_0}$ of the additive group $F.$ Let $P=MN$ denote the Siegel parabolic subgroup of the symplectic group $Sp_{4}(F)$ and let $\rho\in {\rm{Irr}}(GL_{2}(F))$ be regarded as a representation of the Levi subgroup $M$ of $P.$ Denote $\pi={\rm{ind}}_P^{Sp_{4}(F)}(\rho).$

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 43 more