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<p>In this paper, we reveal a submodule relation among the standard modules of <i>p</i>-adic general linear groups. This observation has several applications. We use it to disprove a conjecture by Lapid and Mínguez, which stated that the unique maximal submodule of a standard module is the sum of certain smaller submodules in the Zelevinsky partial order. We also apply a fundamental lemma by Borel–Wallach to investigate the partial order structure of standard modules under the Zelevinsky partial order.</p>

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A Remark on Bernstein–Zelevinsky Classification for General Linear Groups

  • 期刊论文

  • 发表日期：

    2026年7月5日

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  • Caihua Luo,
  • Junwei Zha

摘要

In this paper, we reveal a submodule relation among the standard modules of p-adic general linear groups. This observation has several applications. We use it to disprove a conjecture by Lapid and Mínguez, which stated that the unique maximal submodule of a standard module is the sum of certain smaller submodules in the Zelevinsky partial order. We also apply a fundamental lemma by Borel–Wallach to investigate the partial order structure of standard modules under the Zelevinsky partial order.