In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schroer condition that occurs in the conjectural characterization of $$\square $$ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

中文翻译：

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关于 $${\text {GL}}_n(F)$$ 表示的抛物线归纳的猜想和结果

1980 年，Zelevinsky 引入了某些交换变体，其不可约分量对具有给定上尖支的非阿基米德局部场上一般线性群的复杂不可约表示进行分类。我们为这些分量的某些三元组制定了几何条件，并推测这些条件与抛物线归纳的不可约性有关。这些条件符合 Geiss-Leclerc-Schroer 条件的精神，该条件出现在 $$\square $$ -不可约表示的推测表征中。我们验证了新猜想的一些特殊情况，并检查几何和表示理论条件是否以各种方式兼容。