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Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$

C. Harshitha, C. G. Venketasubramanian

Abstract

Let $F$ be a non-archimedean local field or a finite field. Let $π$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $π_{N,ψ}$ of $π$ with respect to $N$ and a non-degenerate character $ψ$ of $N.$ We also provide a necessary and sufficient condition for $π_{N,ψ}$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $GL_{2n}(F)$ and establish the existence of their Shalika model in the non-archimedean case. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module.

Structure of twisted Jacquet modules of principal series representations of $GL_{2n}(F)$

Abstract

Let be a non-archimedean local field or a finite field. Let be a principal series representation of induced from any of its maximal standard parabolic subgroups. Let be the unipotent radical of the maximal parabolic subgroup of corresponding to the partition In this article, we describe the structure of the twisted Jacquet module of with respect to and a non-degenerate character of We also provide a necessary and sufficient condition for to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of and establish the existence of their Shalika model in the non-archimedean case. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of with a non-zero twisted Jacquet module.
Paper Structure (18 sections, 11 theorems, 26 equations)

This paper contains 18 sections, 11 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Summary of main results
  4. Conjecture on non-vanishing of $\pi_{N,\psi}$
  5. Outline of the article
  6. Few Remarks
  7. Notations and Preliminaries
  8. Notations
  9. Some generalities
  10. Module of an automorphism
  11. Induction functors
  12. Twisted Jacquet module
  13. Some standard subgroups of $G_n$
  14. The character $\psi$ of $N$ and associated subgroups
  15. Preliminary Results
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\rho\in {\rm{Alg}}(M_{r,2n-r})$ and let $\pi$ denote the normalized parabolically induced representation $i_{P_{r,2n-r}}^{G_{2n}}(\rho).$ Then, $\pi_{N,\psi}$ is non-zero if and only if there exists an integer $k$ with $\max \{0,r-n\} \leq k \leq \lfloor \frac{r}{2} \rfloor$ satisfying $\rho_{ and $\psi'_k$ is the character of $N_k'$ defined by $\psi'_k(n') =\psi_0({\rm tr}(x)+{\rm tr}(z))$

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8: D. Prasad
  • Lemma 2.1
  • proof
  • ...and 3 more