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Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments III: properties of minimal sequences

Kei Yuen Chan

TL;DR

The paper analyzes minimal sequences of Bernstein-Zelevinsky derivatives for GL_n(F) via multisegments, establishing a robust combinatorial framework that reveals commutativity and stability properties of minimal sequences. Central tools include the removal process, the η-invariant, and the Zelevinsky ordering, which together yield results such as D_{ rak n- rak n'}(π) ≅ D_{ rak n}(π) when π is minimal to a quotient, and the preservation of minimality under taking submultisegments. The work also develops and applies a realization theorem for highest derivative multisegments, connects these combinatorics to representation-theoretic constructs like embedding models, and proposes conjectural interpretations linking minimality to deeper module-theoretic structures. Together, these results advance the construction of simple quotients of Bernstein-Zelevinsky derivatives and illuminate branching and embedding phenomena in p-adic GL_n representation theory, with potential geometric and categorical extensions.

Abstract

Let $F$ be a non-Archimedean local field. For an irreducible smooth representation $π$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, one associates a simple quotient $D_{\mathfrak m}(π)$ of a Bernstein-Zelevinsky derivative of $π$. In the preceding article, we showed that \[ \mathcal S(π, τ) :=\left\{ \mathfrak m : D_{\mathfrak m}(π)\cong τ\right\} , \] has a unique minimal element under the Zelevinsky ordering, where $\mathfrak m$ runs for all multisegments. The main result of this article includes commutativity and subsequent property of the minimal sequence. At the end of this article, we conjecture some module structure arising from the minimality.

Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments III: properties of minimal sequences

TL;DR

The paper analyzes minimal sequences of Bernstein-Zelevinsky derivatives for GL_n(F) via multisegments, establishing a robust combinatorial framework that reveals commutativity and stability properties of minimal sequences. Central tools include the removal process, the η-invariant, and the Zelevinsky ordering, which together yield results such as D_{ rak n- rak n'}(π) ≅ D_{ rak n}(π) when π is minimal to a quotient, and the preservation of minimality under taking submultisegments. The work also develops and applies a realization theorem for highest derivative multisegments, connects these combinatorics to representation-theoretic constructs like embedding models, and proposes conjectural interpretations linking minimality to deeper module-theoretic structures. Together, these results advance the construction of simple quotients of Bernstein-Zelevinsky derivatives and illuminate branching and embedding phenomena in p-adic GL_n representation theory, with potential geometric and categorical extensions.

Abstract

Let be a non-Archimedean local field. For an irreducible smooth representation of and a multisegment , one associates a simple quotient of a Bernstein-Zelevinsky derivative of . In the preceding article, we showed that has a unique minimal element under the Zelevinsky ordering, where runs for all multisegments. The main result of this article includes commutativity and subsequent property of the minimal sequence. At the end of this article, we conjecture some module structure arising from the minimality.
Paper Structure (65 sections, 53 theorems, 147 equations)

This paper contains 65 sections, 53 theorems, 147 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. The poset $\mathcal{S}(\pi, \tau)$ and the minimal sequence
  4. Main results
  5. Discussions on applications
  6. Organization
  7. Acknowledgements
  8. Highest derivative multisegments
  9. Highest derivative multisegments
  10. Realization Theorem
  11. Combinatorial aspects
  12. Removal processes
  13. More notations on multisegments
  14. Removal process
  15. Properties of removal process
  16. ...and 50 more sections

Key Result

Theorem 1.1

Ch22+e We use the notatons above. Let $\mathfrak n_1, \mathfrak n_2 \in \mathcal{S}(\pi, \tau)$ with $\mathfrak n_1\leq_Z \mathfrak n_2$. If $\mathfrak n \in \mathrm{Mult}_{\rho}$ with $\mathfrak n_1 \leq_Z \mathfrak n \leq_Z \mathfrak n_2$, then $\mathfrak n \in \mathcal{S}(\pi, \tau)$.

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 91 more