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Distinguished Representations with respect to Symmetric Subgroups of $GL_{n}(\mathbb{F}_{q})$

Guy Kapon

TL;DR

This work proves that for $G=GL_{n}(\oldsymbol{F}_{q^{m}})$ with $m\in\{1,2\}$ and a $\mathbb{F}_{q}$-defined involution $\sigma$, every irreducible representation $\pi$ of $G$ that is $H$-distinguished (with $H=G^{\sigma}$) satisfies $\pi\cong\pi^{*,\sigma}$, providing a finite-field analogue of a Prasad–Lapid-type conjecture. The authors leverage Deligne–Lusztig induction for cuspidals, their interaction with restriction of scalars, and Zelevinsky’s parabolic induction framework, encapsulated in an induction lemma that transfers distinguishedness along Levi subgroups. The cuspidal case is handled via Lusztig’s theory of $R_{T,\theta}$ with anisotropic $T$ and general-position $\theta$, and the general case follows by decomposing representations into cuspidal blocks and tracking their $\sigma$-type invariance. The result extends the Prasad–Lapid conjecture to all symmetric subgroups of $GL_{n}$ over finite fields and clarifies the structure of distinguished representations in this setting.

Abstract

We study representations of $GL_{n}(\mathbb{F}_{q})$ that are distinguished with respect to a symmetric subgroup $H=GL_{n}(\mathbb{F}_{q})^σ$, where $σ$ is an involution. We prove that those representations satisfy $π\cong π^{*,σ}$, thus positively answering a version of the Prasad-Lapid conjecture.

Distinguished Representations with respect to Symmetric Subgroups of $GL_{n}(\mathbb{F}_{q})$

TL;DR

This work proves that for with and a -defined involution , every irreducible representation of that is -distinguished (with ) satisfies , providing a finite-field analogue of a Prasad–Lapid-type conjecture. The authors leverage Deligne–Lusztig induction for cuspidals, their interaction with restriction of scalars, and Zelevinsky’s parabolic induction framework, encapsulated in an induction lemma that transfers distinguishedness along Levi subgroups. The cuspidal case is handled via Lusztig’s theory of with anisotropic and general-position , and the general case follows by decomposing representations into cuspidal blocks and tracking their -type invariance. The result extends the Prasad–Lapid conjecture to all symmetric subgroups of over finite fields and clarifies the structure of distinguished representations in this setting.

Abstract

We study representations of that are distinguished with respect to a symmetric subgroup , where is an involution. We prove that those representations satisfy , thus positively answering a version of the Prasad-Lapid conjecture.
Paper Structure (14 sections, 19 theorems, 26 equations)

This paper contains 14 sections, 19 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Ideas of The Proof
  4. Structure of The Paper
  5. Acknowledgments
  6. Preliminaries
  7. Involutions of $GL_{n}$
  8. Deligne-Lusztig
  9. Zelevinsky Theory
  10. The Cuspidal Case
  11. Interaction with Restriction of Scalars
  12. Cuspidal Representations of $GL_{n}$
  13. Induction Lemma
  14. Proof of Main Theorem

Key Result

Theorem A

Let $G=Gl_{n}(\mathbb{F}_{q^{m}})$ where $m$ is either $1$ or $2$, let $\sigma$ be an algebraic involution, defined over $\mathbb{F}_{q}$, let $H=G^{\sigma}$ be a symmetric subgroup, and let $\pi \in Irr(G)$ be an irreducible representation. If $\pi^{H}\neq 0$ then $\pi \cong \pi^{*}\circ \sigma$.

Theorems & Definitions (41)

  • Conjecture 1.1
  • Theorem A
  • Theorem 1.2: Lusztig2007
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 2.1
  • Definition 2.2
  • ...and 31 more