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On the product functor on inner forms of the general linear group over a non-Archimedean local field

Kei Yuen Chan

Abstract

Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $σ$ of $G_n$. Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $π\boxtimes σ$ when $π$ is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some $G_m$, on which the parabolic induction functor $τ\mapsto τ\times σ$ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.

On the product functor on inner forms of the general linear group over a non-Archimedean local field

Abstract

Let be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation of . Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form when is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some , on which the parabolic induction functor is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.
Paper Structure (44 sections, 49 theorems, 153 equations)

This paper contains 44 sections, 49 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Notations
  4. Basic notations
  5. More notations for segments and multisegments
  6. Langlands and Zelevinsky classification
  7. Parabolic inductions and Jacquet functors
  8. Geometric lemma
  9. Jacquet functors on segment and Steinberg representations
  10. Some generalities of the product functor
  11. Product functor
  12. Smooth dual functor
  13. Cohomological dual functor
  14. Product with a segment representation and intersection-union process
  15. Some explicit criteria for a multisegment in $\mathcal{M}_{\pi}$
  16. ...and 29 more sections

Key Result

Theorem 1.1

(=Theorem thm intersection-union closed) Let $\pi$ be an irreducible representation of $G_n$. For $\mathfrak n \in \mathcal{M}_{\pi}$, if $\mathfrak n'$ is another multisegment with $\mathfrak n' \leq_Z \mathfrak n$, then $\mathfrak n' \in \mathcal{M}_{\pi}$.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 88 more