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Study of multiplicities in induced representations of \(GL_n\) through a symmetric reduction

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Abstract

In this article, we develop a process to symmetrize the irreducible admissible representations of \(GL_n(F)\) with F being a finite extension of \(\mathbb {Q}_p\), as a consequence we obtain a more geometric understanding of the coefficient \(m(\mathbf {b}, \mathbf {a})\) appearing in the decomposition of parabolic inductions, which allows us to prove a conjecture inspired by Zelevinsky.

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Acknowledgements

This paper is part of my thesis at University Paris 13, which is funded by the program DIM of the region Ile de France. I would like to thank my advisor Pascal Boyer for his keen interest in this work and his continuing support and countless advice. It is rewritten during my first year of posdoc at Universität Bonn, I thank their hospitality. In addition, I would like to thank Alberto Mínguez et Vincent Sécherre, Yichao Tian for their helpful discussions on the subject. Finally, I would like to thank an anonymous referee for a careful reading and many suggestions for a previous version of the paper.

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  1. Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, China

    Taiwang Deng

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  1. Taiwang Deng
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Correspondence to Taiwang Deng.

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Deng, T. Study of multiplicities in induced representations of \(GL_n\) through a symmetric reduction. manuscripta math. 171, 23–72 (2023). https://doi.org/10.1007/s00229-022-01375-1

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